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ACTUARIAL  STUDIES  NO.  2

CONSTRUCTION  OF  MORTALITY  TABLES
FROM  THE  RECORDS  OF  INSURED  LIVES

t      ^^RINCIPAL  CONTRIBUTORS  j.

RAY  D.  MURPHY  AND  PERCY  C.  Hf  PAPPS

PUBLISHED   BY

THE  ACTUARIAL  SOCIETY  OF  AMERICA

1922

ORIGINAL  COMMITTEE
Arthur  Hunter  Wendell  M.  Strong

Henry  Moir  A.  A.  Welch

P.  C.  H.  Papps  a.  B.  Wood

John  K.  Gore,  Chairman

COMMITTEE  NOW  IN  CHARGE

Robert  Henderson  Wendell  M.  Strong

J.  M.  Laird  J.  S.  Thompson

A.  T.  Maclean  Hugh  H.  Wolfenden

A.  H.  Mowbray  Henry  Moir,  Chairman

Thb  Actuarial  Societt  of  America

New  York

PRESS  OF

THC  NEW  EIM  PRINTINQ  COMPANY

LANCASTER,  PA.

CONSTRUCTION  OF  MORTALITY  TABLES  FROM  THE
RECORDS   OF  INSURED  LIVES.

Chapter  I.     Introduction.

A  mortality  table  starts  with  a  group  of  persons  at  a  specified
age  and  shows  the  number  of  survivors  at  each  subsequent  age.
There  is  also  generally  set  down  the  number  dying  in  each  year
of  age.  The  radix  of  the  table,  or  the  number  living  at  the
youngest  age  shown,  is  arbitrarily  selected.  Then,  from  the
values  of  px  for  each  age,  the  values  of  h  may  be  computed
successively  for  the  higher  ages  by  the  relation,  Upx  =  h+i*

As  a  general  rule  the  construction  of  a  mortality  table  based
on  the  records  of  insured  lives  is  effected  by  ascertaining  the
value  of  5x  for  each  a^e.  This  vahie  is  obtained  by  dividing
6x  by  Ex,  where  6x  represents  the  deaths  and  Ex  the  exposed  to
risk  of  death  for  the  year  between  ages  x  and  x  +  1  in  the  data
under  observation.!

In  practice  some  persons  will  be  found  who  are  under  observa-
tion for  only  a  part  of  a  given  year  of  age  either  because  they
enter  the  experience  after  the  beginning  of  the  year  or  because
they  pass  out  of  it  for  causes  other  than  death  before  the  end
of  the  year.  In  such  cases  each  person  is  counted  in  computing
Ex  as  a  fraction  equal  to  the  proportion  of  the  year  under  ob-
servation. The  degree  of  accuracy  with  which  such  fractions  are
computed  varies  according  to  the  method  used  in  tabulating  the
data.  A  person  who  dies  between  ages  x  and  x  +  1  must  be
included  in  Ex  as  exposed  to  risk  for  the  full  year  and  not  a
fraction,  because  Qx  represents  the  proportion  of  Ix  persons  alive
at  age  x  who  will  not  be  alive  at  age  x  -\-  I.

*  See  Institute  of  Actuaries'  Text  Book,  Part  II,  Chapter  I.

t  As  will  be  seen  in  Chapter  III,  6x  may  be  the  number  of  deaths  and  Eg
the  number  of  Uves  exposed  to  risk  of  death,  or  they  may  represent  respectively
the  mmiber  of  policies  terminated  by  death  and  the  number  exposed  to  risk
of  termination  by  death;  or  lastly  they  may  be  the  amoimt  of  insurance
terminated  by  death  and  the  amount  exposed  to  risk.  It  is  important  to
note,  however,  that  both  numerator  and  denominator  in  the  fraction  dx/E,
must  always  relate  to  the  same  kind  of  data.

1

2  CONSTRUCTION    OF    MORTALITY   TABLES

It  should  be  remembered  that  theoretically  the  numerical
value  of  the  radix  selected  and  the  resulting  size  of  the  values  of
Ix  and  dx  are  of  no  significance,  but  that  the  relative  values  of
these  functions  are  of  vital  importance.  The  use  of  a  large  radix
is  advisable,  however,  so  that  when  the  calculated  values  of
Ix  and  dx  are  adjusted  to  the  nearest  integer  the  necessary  error
introduced  is  insignificant.  This  will  also  have  a  bearing  on  the
limiting  age  (w)  since  this  would  be  the  lowest  age  for  which  the
unadjusted  value  of  h  is  less  than  .5,  unless  fractions  are  shown.

When  an  applicant  is  accepted  for  insurance  after  being  ex-
amined by  the  company's  physicians,  he  is  a  "select"  life.
Among  a  number  of  such  lives  after  the  lapse  of  a  few  years  there
will  be  some  whose  health  has  become  impaired  to  a  greater  or
less  degree,  while  others  will  remain  as  healthy  as  when  first
examined  for  insurance.  The  survivors  of  a  body  of  select  lives
are  therefore  called  "mixed"  lives.

It  follows  that  the  rate  of  mortality  of  insured  persons  of  a
given  attained  age,  say  x,  will  not  be  the  same  among  persons
just  insured  at  age  x  as  among  persons  insured  n  years  ago  at
age  X  —  n.  It  has  been  found  by  experience  that  persons  just
insured  at  age  x  are  subject  to  a  lower  rate  of  mortality  than
those  also  aged  x,  but  insured  at  age  x  —  1.  These  latter,  in
turn,  usually  show  a  lower  rate  of  mortality  than  those  aged  x,
but  insured  at  age  x  —  2.  In  general,  for  limited  values  of  n,
which  vary  in  different  experiences  and  in  different  age  sections
of  a  single  experience,  it  is  found  that  g[x_^in')+;nT  <  9[x-ni+n
where  the  portion  of  the  suffix  within  the  square  brackets  in-
dicates the  age  at  issue,  and  the  other  portion,  the  duration  since
entry,  the  total  being  the  present  age,  i.e.,  x.

As  n  increases,  the  extent  of  the  difference  will  be  found  to
decrease,  so  that  if  g[i_;izi]+^:ii  =  ?[*-n]fn  —  5„  then  as  n  in-
creases, 5„  will  approach  the  limit  zero.

If  t  be  the  greatest  value  of  n  for  which  the  relation  ^[x-iiZii+s:!^^
<  9[i_„]+n  holds,  this  fact  is  expressed  by  saying  that  the  effects
of  selection  last  for  t  years.  Accidental  fluctuations  in  the  data,
on  which  the  mortality  rates  are  based,  are  alone  sufficient  to
prevent  any  exact  determination  of  the  value  of  t.  An  approxi-
mate value  is  all  that  can  be  expected.  In  practice,  t  may  have
a  small  value,  as,  for  instance,  in  the  case  of  residents  in  the
tropics,  where  values  as  low  as  2  years  or  even  1  year  may  be
found;  or  it  may  have  such  a  comparatively  large  value  as  10

FROM   THE   RECORDS   OF  INSURED    LIVES.  6

years  or  more,  the  latter  figure  applying  in  the  case  of  the  British
Offices'  Experience  (1863-1893)  under  whole  life  participating
policies,  especially  at  the  younger  ages.  It  may  be  questioned
whether  the  effects  of  selection  ever  entirely  disappear  or  whether
they  become  so  merged  with  other  influences,  such  as  changes  in
sanitary  conditions  and  in  the  mortality  of  the  general  popula-
tion, that  they  are  lost.*  As  a  practical  matter,  however,  we
are  warranted  in  assuming  that  they  cease  after  a  certain  period.

Now,  if  a  body  of  select  lives  all  of  a  given  age  be  observed,
and  the  rate  of  mortality  resulting  during  the  first,  second,
third,  etc.,  years  of  insurance  be  set  forth  for  each  year,  the
result  will  be  a  select  table  of  mortality  for  that  particular  age
at  entry.  If  similar  tables  be  prepared  for  each  age  at  date  of
selection,  we  obtain  a  set  of  "select  mortality  tables."

It  would  involve  much  labor,  however,  to  base  calculations  on
such  a  set  of  select  tables.  A  trial  is  therefore  made  to  ascertain
the  effective  period  of  selection  beyond  which  the  rate  of  mor-
tality appears  to  depend  only  upon  the  attained  age  and  may
consequently  be  formed  into  one  "ultimate"  table.  This  may
be  done  by  a  direct  comparison  of  the  values  found  for  qix-n]+n
for  different  values  of  n  as  indicated  above.  Such  a  comparison
may  be  confusing,  however,  because  of  the  large  number  of
values  to  be  observed  and  the  fluctuations  in  them,  and  it  will
usually  be  more  satisfactory  to  determine  by  observation  ap-
proximately where  the  line  of  division  lies  and  then  apply  a
different  final  test.

In  describing  the  construction  of  the  American  Men  Tables,  it
was  stated  that  "the  crude  death  rates  were  deduced  for  each
of  the  first  five  insurance  years,  for  the  sixth  and  succeeding
years  combined,  and  for  the  eleventh  and  succeeding  years
combined.  The  expected  deaths  for  each  of  the  first  ten  in-
surance years  were  then  calculated  by  graded  rates  of  mortality
based  upon  the  data  for  the  sixth  and  succeeding  insurance  years
in  order  that  the  number  of  years  for  which  medical  selection
lasted  could  be  determined.  It  was  seen  that  the  material  for
the  sixth  and  succeeding  insurance  years  could  be  safely  com-
bined according  to  attained  age."

A  different  method  was  used  in  compiling  the  O^^^  table.  The
expectations  of  life  were  employed,  as  they  would  not  be  subject

*  See  T.A.S.  A.,  Vol.  XIII,  page  211,  for  a  discussion  of  the  effect  on  select
tables  of  a  variation  in  mortality  during  the  period  of  investigation.

CONSTRUCTION   OF   MORTALITY  TABLES

to  fluctuations  to  the  same  degree  as  would  the  mortality  rates
for  individual  ages.  On  page  146  of  "Account  of  Principles  and
Methods"  of  that  experience  are  shown,  for  quinquennial  groups
of  ages,  the  values  of  the  expectations  eix],  e[x_5]+5,  e[i_io]+io,  etc.,
to  e[x_2B]+75.  There  are  also  given  the  values  of  ej-^^  and  ex^^*^^}
the  expectations  of  life  found  by  combining  the  data  for  the  same
attained  age,  but  excluding  the  data  for  the  first  five  and  first
ten  years  of  duration  respectively.  If  selection  were  still  effect-
tive  in  the  (n  +  l)th  year,  e[i_„]+„  would  be  greater  than  Cx^"^.
It  was  decided  for  practical  advantages  to  consider  that  the  effect
of  selection  had  disappeared  in  ten  years,  although  this  did  not
appear  to  be  true  for  all  ages  at  entry.

A  select  and  ultimate  table  may  be  set  forth  conveniently  as
shown  by  the  following  section  of  the  O"^^'^^  table.

Age  at
Entry.

Years  Elapsed  Since  Date  of  Insurance.

Age  At-
tained.

.    0

1

2

3

4

6  or
more.

X.

Im-

hx]+i-

hx]+t-

l[x]+i.

^(x]+4.

llx]+t.

1+5.

20
21
22
23
24
25
26
27
28
29
30
31
32
33

100,000
99,264
98,530
97,794
97,055
96,316
95,567
94,818
94,059
93,300
92,529

99,580
98,844
98,109
97,369
96,630
95,887
95,135
94,382
93,618
92,854

99,003
98,267
97,530
96,790
96,048
95,302
94,547
93,791
93,023

98,333
97,596
96,857
96,115
95,370
94,619
93,862
93,100

97,616
96,877
96,135
95,389
94,639
93,884
93,122

96,879
96,137
95,392
94,641
93,886
93,124

25
26
27
28
29
30
31
32
33
34
35
36
37
38

When  the  rates  of  mortality  are  obtained,  the  first  line  of  the
table  may  be  started  with  the  desired  radix  and  the  values  suc-
cessively computed  across  the  first  line  and  then  down  the  last
column.  The  second  and  subsequent  lines  may  be  calculated
by  working  back  from  the  ultimate  column  by  means  of  the
equality  log  lix]+n-i  =  log  lix]+n  —  log  p^+n-i-

If  the  data  entering  into  both  the  select  and  ultimate  sections
of  the  table  be  combined,  or,  in  other  words,  if  a  mortality  table
be  formed  according  to  age  only,  irrespective  of  the  year  of
insurance,  the  result  will  be  an  "aggregate  table."

FROM   THE   RECORDS   OF   INSURED   LIVES.  5

While  it  may  be  known  that  the  effects  of  selection  last  for
several  years,  it  may  be  thought  desirable  for  practical  purposes
to  construct  a  table  of  mortality  excluding  the  experience  of,
say,  the  first  two  years  only,  without  constructing  the  select
tables  corresponding  to  those  first  two  years.  Such  a  table  is
known  as  a  "truncated"  table.  Every  ultimate  table  is  in  a
sense  a  truncated  table,  but  the  name  "ultimate"  is  usually
applied  only  to  a  table  which  forms  the  continuation  of  the
select  section  of  a  mortality  table.  In  the  select  section  the
rate  of  mortality  is  shown  for  the  age  attained,  but  modified
according  to  the  length  of  time  elapsed  after  initial  selection.
In  an  aggregate  or  an  ultimate  table  the  rate  of  mortality  is
shown  for  each  age  attained,  without  modification.

It  is  desirable  to  consider  the  effects  of  the  duration  of  in-
surance of  the  data  entering  into  an  aggregate  table.  It  will
be  understood  that  the  following  remarks  will  apply  generally
but  in  a  modified  degree  to  a  truncated  table.  By  first  obtaining
a  clear  idea  of  the  nature  of  these  different  forms  of  mortality
tables,  the  student  will  be  in  a  better  position  to  grasp  the  sig-
nificance of  the  various  methods  of  investigating  and  collecting
the  data.

For  the  sake  of  illustration,  let  it  be  first  assumed  that  at  all
ages  the  effects  of  selection  will  last  for  ten  years  only.  Then,
if  an  aggregate  table  be  formed  from  the  experience  of  a  company
that  has  been  in  business  for  only  ten  years,  the  resulting  table
will  be  composed  only  of  lives  which  have  not  reached  the  ultimate
rates  of  mortality.  The  rates  of  mortality  shown  will  obviously
be  much  less  than  will  be  the  case  when  the  table  is  based  upon
the  total  experience  of  an  old  company  which  has  been  many
years  in  business;  for  in  the  latter  case,  the  higher  ultimate
rates  of  mortality  of  the  old  business  will  be  included,  raising
the  aggregate  rates  for  any  given  age  above  the  lower  mortality
of  the  newer  business  for  the  same  age.  Again,  if  there  be  two
companies  of  the  same  age,  the  aggregate  tables  formed  from  the
experience  of  the  respective  companies  will  differ  considerably
if  one  company  has  recently  been  writing  a  much  larger  business
in  relation  to  its  size  than  the  other.  The  company  having  the
larger  proportion  of  select  business  in  its  aggregate  data  will,
other  conditions  being  the  same,  show  the  lower  mortality  ex-
perience. In  making  this  statement  it  is  assumed  that  the  age
distribution  is  similar  and  that  the  companies  are  subject  to

6  CONSTRUCTION    OF    MORTALITY    TABLES.

the  same  select  rates  of  mortality,  the  difference  in  the  aggregate
mortality  being  due  solely  to  the  different  proportions  of  new  and

From  what  has  been  said  it  will  be  clear  that  a  select  table  is
The  great  convenience  of  an  aggregate  or  an  ultimate  table  for
actuarial  calculations  however  makes  the  use  of  such  a  table
desirable  if  it  can  be  shown  to  produce  adequate  results.  The
Combined  Experience,  the  H^  and  O^  Tables  are  aggregate  tables,
the  American  Experience  an  ultimate,  and  O^^*^  a  truncated
aggregate  table.  These  tables  have  served  a  very  useful  purpose
as  the  basis  of  insurance  premiums  and  reserves.*

For  a  basis  of  comparison  in  mortality  investigations  select
tables  are  essential  to  prevent  erroneous  conclusions  due  to  the
difference  in  the  average  policy  duration  of  two  or  more  classes
of  policyholders.

*  See  T.  A.  S.  A.,  Vol.  XII,  page  49,  for  application  of  the  principle  of
truncated  tables.

Chapter  II .    Sources  of  Data.

Any  records  showing  the  distribution  according  to  ages  or
groups  of  ages  of  persons  living  and  the  number  of  deaths  oc-
curring among  such  persons,  may  be  used  for  the  construction
of  a  mortality  table,  provided  there  are  a  sufficient  number  under
observation  to  permit  the  law  of  average  to  operate  and  to  show
the  general  trend  of  the  mortality.  The  larger  the  number  of
lives  involved,  the  more  reliable,  as  a  general  rule,  will  be  the
resulting  mortality  table.  While  it  is  preferable  to  be  able  to
ascertain  the  exposed  to  risk  and  the  deaths  for  each  age,  it  is
not  essential;  for  if  the  average  values  of  qx  for  groups  of  ages  are
known,  a  mortality  table  may  be  constructed  by  interpolation
showing  the  values  of  qx  for  every  age.*

Population  statistics  have  been  utilized  for  compiling  mor-
tality tables.  With  the  properly  compiled  records  of  two  censuses
and  of  the  intervening  deaths  a  reliable  table  may  be  constructed.
The  Carlisle  Table  and  the  English  Life  Tables  are  examples  of
tables  which  on  the  whole  are  properly  constructed,  while  the
Northampton  Table  is  an  example  of  an  unsuccessful  attempt  to
construct  a  mortality  table  from  the  records  of  deaths  only.
The  construction  of  a  mortality  table  from  population  statistics
is  covered  in  No.  3  of  Actuarial  Studies,  and  the  characteristics
of  the  tables  mentioned  are  set  forth  in  No.  1.

The  records  of  insurance  companies  and  fraternal  societies
form  the  most  valuable  source  for  compiling  mortality  tables
for  the  use  of  such  companies  and  societies.  The  tables  are
required  for  measuring  the  mortality  among  practically  the  same
class  of  lives  as  that  on  which  the  tables  are  based.  The  Amer-
ican Experience  Table  may  be  cited  as  one  based  on  the  experi-
ence of  an  American  life  insurance  company,  namely.  The
Mutual  Life  Insurance  Company  of  New  York.  The  American
Men  Table  was  formed  from  the  combined  data  of  many  Amer-
ican and  Canadian  companies  showing  the  experience  on  Amer-
ican males.  The  Combined  Experience,  or,  as  it  is  sometimes
called,  the  Actuaries',  or  Seventeen  Offices'  Table,  the  H^  Table
and  the  0^  Table  are  based  on  the  experience  of  groups    of

*  See  Actuarial  Studies  No.  4.

7

8  CONSTRUCTION    OF   MORTALITY   TABLES.

British  companies.  The  experience  of  a  number  of  fraternal
societies  was  used  as  the  basis  of  the  National  Fraternal  Congress
Table.  These  tables  are  described  in  No.  1  of  Actuarial  Studies.
Among  the  various  miscellaneous  sources  from  which  mortality
tables  may  be  compiled  are  the  volumes,  published  annually,
giving  details  concerning  the  families  of  the  British  Peerage.
These  volumes  supply  data  relating  to  British  Peers,  the  sons
and  daughters  of  Peers  and  the  sons  and  daughters  of  the  eldest
sons  of  Peers,  from  which  a  number  of  Peerage  Tables  of  mor-
tality have  been  constructed.  The  records  kept  by  universities
have  also  been  employed  in  compiling  a  mortality  table.  From
the  records  of  Widows'  and  Pension  Funds  tables  of  mortality
may  be  formed  which  will  be  useful  as  a  guide  to  the  management
of  such  funds.

Chapter  III.    Methods  of  Recording  the  Data.

As  this  study  deals  only  with  mortality  tables  from  the
records  of  insured  lives  the  methods  hereafter  described  will  be
those  applicable  to  the  data  available  in  an  insurance  company
or  fraternal  societJ^  The  facts  regarding  each  policy  will  be
set  down  on  a  card  so  that  they  may  be  sorted  and  tabulated
(See  Chapter  IV).  The  details  to  be  recorded  will  depend  partly
upon  the  use  for  which  the  mortality  table  is  desired,  and  methods
may  be  varied  to  facilitate  the  work,  provided  accuracy  is  not
thereby  sacrificed.

There  are  three  possible  bases  for  determining  5^,  (1)  the  death
rates  among  the  lives  insured,  (2)  the  rates  of  termination  by
death  of  policies  in  force,  and  (3)  the  rates  of  termination  by
death  of  the  amounts  insured.

If  it  is  desired  to  investigate  by  lives,  the  data  concerning  all
the  policies  on  each  life  must  be  brought  together  to  avoid
duplication.  Where  the  experience  is  confined  to  one  company
the  office  records  will  usually  show  all  policies  on  a  single  life,
but  when  the  combined  experience  of  several  offices  is  under
investigation,  the  existence  of  duplicate  policies  in  different
offices  can  be  ascertained  only  by  examination.  The  usual  plan
is  to  arrange  all  names  in  strict  alphabetical  order — as  in  a
directory — and  thus  enable  the  bulk  of  the  duplicates  to  be
brought  together.  Where  names  agree,  but  the  dates  of  birth
differ  radically,  different  lives  may  be  presumed ;  where  the  dates
of  birth  differ  slightly,  it  may  sometimes  be  found  that  the  cards
refer  to  a  single  life.  Occasionally  day  and  month  of  birth  will
agree,  but  not  the  year.  Inquiry  as  to  occupation  and  residence
will  usually  throw  light  upon  doubtful  cases.

Some  persons,  however,  are  careless  in  the  use  of  their  names,
and  it  will  occasionally  happen  that  a  name  may  appear  as,  say,
George  Frederick  Smith  in  the  records  of  one  company,  and
Frederick  Smith  in  those  of  another — or  the  surname  may  be
written  Smyth.  Alphabetical  sorting  will  fail  to  disclose  the
existence  of  such  duplicates,  and,  where  complete  elimination  is
considered  important,  an  independent  sorting  of  the  material
by  dates  of  birth  should  be  effected.     This  latter  device  is

9

10  CONSTRUCTION    OF   MORTALITY   TABLES

especially  useful  in  the  case  of  female  lives  where  change  of  name
owing  to  marriage  is  quite  often  recorded  in  one  office  and  not
in  another,  owing  to  carelessness  on  the  part  of  the  insured,  or
to  the  termination  of  a  policy  prior  to  marriage.  Even  so,
errors  in  dates  of  birth  will  prevent  complete  elimination  of
duplicates.

Having  brought  the  duplicates  together  the  method  of  elimina-
tion depends  upon  the  kind  of  table  that  is  to  be  constructed.
If  it  is  an  aggregate  table  and  there  has  been  a  continuous  ex-
posure from  the  date  of  entry  of  the  earliest  policy  to  the  last
date  of  exit,  one  card  may  be  written  for  the  complete  exposure
and  the  others  destroyed.  If  there  has  not  been  a  continuous
exposure  then  two  or  more  cards  must  be  written  to  cover  the
several  continuous  periods  that  the  company  was  at  risk.  If
select  tables  are  to  be  compiled  the  life  should  count  only  once
in  the  ultimate  section  of  the  table,  but  may  be  counted  once  for
each  select  period  beginning  at  a  different  age.  This  is  seen  from
the  fact  that  each  such  select  period  is  a  separate  experience  and
it  involves  no  duplication  to  allow  one  life  to  enter  two  or  more
such  experiences.

A  difficulty  enters  here  because  the  duration  of  the  select
period  cannot  be  determined  in  advance.  One  card  may  there-
fore be  written  for  the  complete  exposure  and  separate  cards  for
the  other  select  periods,  using  an  assumed  duration  for  the  period
of  selection.  These  latter  cards  should  be  coded  so  that  they
may  subsequently  be  sorted  out  and  the  durations  corrected
after  the  select  period  has  been  decided  upon.

The  British  Offices'  Experience  illustrates  the  practical  diffi-
culties which  may  arise  in  eliminating  duplicates.  Several
sections  of  the  experience  were  to  be  compiled  independently,
combining  the  data  in  some  of  these  afterward.  Furthermore
aggregate,  truncated  and  select  tables  had  to  be  provided  for.
Strict  elimination  in  every  case  would  have  been  too  laborious
and  consequently  duplication  was  considered  in  each  section
only.  When  sections  were  combined  some  lives  were  therefore
recorded  more  than  once.  Furthermore  one  card  could  not
well  be  written  for  the  continuous  exposure  on  one  life  for  the
aggregate  table,  as  this  would  conceal  the  select  experience
arising  from  policies  issued  after  the  earliest  one  which  should
properly  enter  the  select  tables,  and  also  would  make  the  trun-
cated tabulation  desired  impossible.     A  card  was  consequently

FROM   THE   RECORDS   OF   INSURED    LIVES.  H

written  for  each  age  at  which  a  policy  was  issued  but  such  cards
were  marked  to  enter  the  aggregate  experience  only  at  the  dura-
tion corresponding  to  the  date  of  exit  of  the  preceding  policy.
The  result  was  that  the  select  tables  contained  duplication  in  the
ultimate  section  arising  from  policies  on  one  life  issued  at  dif-
ferent ages.  The  truncated  table  likewise  contained  two  or
more  coincident  exposures  on  some  lives.  The  investigation  was
therefore  not  based  wholly  on  lives  but  on  a  combination  of
lives  and  policies.*

Mr.  G.  F.  Hardy  has  stated,  in  "The  Theory  of  the  Construc-
tion of  Tables  of  Mortality,  etc.,"  p.  18:

"Without  dogmatizing  upon  the  point,  it  appears  to  me  that  the  proper
course  is,  where  two  or  more  policies  are  effected  at  the  same  time  or  at  the
same  age  at  entry,  to  treat  them  as  a  single  risk,  but  where  the  subsequent
poUcies  are  effected  at  later  ages,  involving  fresh  medical  selection,  to  treat
them  as  separate  risks.  This  means  the  elimination  of  duphcates  in  each  of
the  'select'  tables  for  individual  ages  at  entry,  but  no  fm-ther  elimination  in
the  resulting  aggregate  tables,  a  course  which  has  the  advantage  of  making  the
aggregate  table  the  true  aggregate  of  the  tables  for  separate  ages  at  entry.
Judging  by  the  results  of  the  0"  experience,  this  course  is  necessary  if  we  are  to
produce  an  aggregate  table,  representing  'ultimate'  rates  of  mortality  after
the  lapse  of  a  stated  period  from  entry,  which  will  join  on  smoothly  to  the
'select'  rates."

If  the  investigation  is  by  policies  there  will  be  cases  where  one
life  is  insured  under  several  policies  and  the  failure  of  one  such
life  will  have  a  greater  effect  upon  the  mortality  table  than  the
failure  of  a  life  insured  under  a  single  policy.  Now,  if  the  lives
which  are  insured  under  several  policies  are  on  the  whole  subject
to  a  lower  rate  of  mortality  than  lives  of  similar  ages  insured
under  single  policies,  it  follows  that  the  table  based  on  policies
will  show  lower  rates  of  mortality  than  if  the  table  had  been
based  on  lives  and  vice  versa.  The  general  effect  of  investigating
by  policies  is  to  produce  rates  of  mortality  fairly  close  to  those
determined  by  lives  insured  except  for  accidental  fluctuations
where  the  exposures  are  small;  but  it  has  been  thought  that
an  experience  by  policies  may  show  a  slight  tendency  to  diminish
the  values  of  q^  at  the  younger  and  increase  them  at  the  older
ages  in  an  aggregate  table.  This  is  probably  due  to  the  fact
that  at  the  younger  ages,  where  some  individuals  are  repeatedly

*  For  an  instructive  detailed  account  of  the  manner  in  which  the  cards  were
marked  and  duphcates  ehminated  see  the  volume  of  the  British  OflGices'  Life
Tables,  1893,  entitled  "Account  of  Principles  and  Methods."

12  CONSTRUCTION   OF  MORTALITY   TABLES

undergoing  medical  examination  to  obtain  additional  policies,
those  who  are  successful  represent  a  class  of  superior  lives,  and
the  additional  weight  given  to  them  in  the  experience  tends  to
lower  the  mortality;  while  those  who  are  unsuccessful  do  not
show  a  proportionate  effect.  At  the  older  ages  where  few  addi-
tional policies  are  taken  this  extra  weight  operates  to  increase  the
proportion  that  the  ultimate  data  bears  to  the  whole.

The  saving  of  labor  in  investigating  by  policies  rather  than  by
lives  will  frequently  justify  the  former  method.  The  result  will
usually  be  a  satisfactory  basis  for  comparison  with  other  mor-
tality tables  and  with  an  insurance  company's  experience  in
special  classes  of  risks.  Neither  of  these  methods  however  is
necessarily  a  safe  basis  for  the  computation  of  premiums  and
reserves,  since  the  rate  of  financial  loss  may  be  greater  than  the
rate  of  mortality  by  lives  or  policies.  This  will  be  the  case
when  the  mortality  among  lives  insured  for  large  amounts  is
greater  than  among  those  insured  for  small  and  moderate
amounts.

Investigations  by  amounts  insured,  in  which  qx  is  determined
as  the  ratio  of  claims  incurred  to  insurance  in  force,  are  con-
sidered essential  under  such  circumstances,  which  are  common
in  this  country.  It  is  evident  that  mortality  rates  so  derived
will  be  subject  to  accidental  fluctuations  caused  by  the  failure
of  one  or  more  lives  insured  for  large  amounts.  Such  fluctua-
tions may  subsequently  be  removed  by  graduation  or  may  be
lessened  by  restricting  the  limit  of  insurance  for  which  any  one
life  will  be  counted  as  exposed  in  the  experience.  This  latter
restriction  would  necessitate  the  labor  of  bringing  together  for
investigation  all  policies  on  any  life  insured  for  a  large  amount.
The  use  of  amounts  insured  introduces  additional  work  also
because  of  the  necessity  for  noting  all  changes  in  amount  after
issue.

In  the  American-Canadian  Mortality  Investigation  (1900-
1915)  it  was  decided  to  count  as  \$100,000  only,  insurance  on  any
individual  life  issued  at  any  one  age  for  more  than  that  sum.  In
order  to  accomplish  this  purpose  there  was  noted  on  the  cards
furnished  by  each  company  the  insured's  initials  and  full  date  of
birth  if  the  policy  was  for  \$50,000  or  more.  For  each  age  at
entry  those  cards  were  brought  together  which  had  the  same
date  of  birth  and  initials.  The  mortality  was  to  be  investigated
also  by  plans  of  insurance  and  Volume  I  of  that  experience

FROM   THE   RECORDS   OF  INSURED   LIVES.  13

states,  "When  more  than  \$100,000  of  insurance  was  issued  at
the  same  entry  age  on  the  same  life  but  on  different  plans  of
insurance,  the  amount  of  insurance  on  each  card  was  reduced
proportionately,  provided  the  mode  of  termination  and  duration
were  identical.  For  instance,  if  \$50,000  had  been  issued  on  the
twenty-payment  life  plan  and  \$100,000  on  the  ordinary  life
plan,  these  amounts  were  reduced  to  \$33,300  and  \$66,700  re-
spectively. This  procedure  made  it  unnecessary  to  make  any
of  insurance.

"When  the  duration  of  the  several  policies  differed,  the
policy  with  the  longest  duration  was  retained  in  the  investigation
for  its  original  amount,  provided  such  amount  did  not  exceed
\$100,000."

This  method  guarded  against  an  undue  effect  during  the
select  period  of  the  failure  of  a  life  insured  for  a  very  large  amount.
There  was  not  a  similar  necessity  for  restricting  the  amount  in
the  ultimate  portion  of  the  tables  because  of  the  larger  volume
of  exposures  during  that  period.  It  will  also  be  noticed  that
the  limit  of  \$100,000  might  be  exceeded  even  during  the  select
period  through  the  issuance  of  policies  for  less  than  \$50,000  at
the  same  age,  but  this  was  not  a  serious  practical  objection  to
the  method  followed.

In  this  investigation  it  was  decided  to  terminate  the  exposure
if  the  amount  insured  under  a  policy  was  increased  or  decreased.
This  eliminated  much  labor  and  avoided  complications  regarding
the  select  period  where  such  a  change  was  made  upon  evidence
of  insurability.

A  life  may  pass  out  of  observation  by  any  one  of  four  modes  of
termination:  (1)  Existing;  (2)  Withdrawn;  (3)  Matured;  (4)
Died.  The  exposure  of  a  life  terminates  by  "existing"  when  the
policy  is  in  existence  at  the  close  of  the  period  covered  by  the
experience.  Lapses  and  surrenders  are  classified  as  withdrawn
All  involuntary  withdrawals  such  as  expired  term  policies  and
matured  endowments  are  treated  as  matured.  The  withdrawn
and  matured  have  often  been  grouped  together,  while  on  the
on  whether  the  experience  is  to  be  used  for  additional  information,
such  as  the  rate  of  voluntary  withdrawal.

In  recording  the  exposures  and  deaths  in  a  mortality  experi-
ence three  methods  are  available;  the  "Policy  Year,"  "Calendar

14  CONSTRUCTION   OF   MORTALITY  TABLES

Year,"  and  "Life  Year"  methods.     These  terms  apply  to  the
manner  of  analyzing  the  exposures  and  deaths.

A.  Policy  Year  Method.

Under  the  policy  year  method  the  exposures  are  traced  from
the  beginning  to  the  end  of  each  policy  year  and  each  death  is
allocated  within  the  exact  policy  year  in  which  it  occurs.  The
age  at  entry  may  be  taken  as  the  age  nearest  birthday,  or  it  may
be  taken  as  the  mean  age,  found  by  subtracting  the  calendar
year  of  birth  from  the  calendar  year  of  issue.  While  in  any
particular  case  the  mean  age  may  be  nearly  one  year  greater  or
less  than  the  correct  age,  these  discrepancies  may  be  considered
to  balance.  In  the  United  States  the  nearest  age  method  can
be  followed  by  extracting  the  age  directly  from  the  policy  records.
In  Great  Britian  and  Canada,  however,  it  is  customary  to  insure
at  the  age  next  birthday  and  either  method  would  require  a
calculation.

As  a  policy  year  investigation  ordinarily  begins  and  ends  with
policy  anniversaries  in  specified  calendar  years  the  existing  will
usually  pass  out  of  observation  at  an  integral  age  determined  by
adding  to  the  age  at  issue  the  duration  found  by  subtracting  the
year  of  issue  from  the  year  with  which  the  experience  closes.

In  treating  the  withdrawn  fractional  durations  are  involved.
These  may  be  treated  as  the  "exact,"  "nearest,"  or  "mean"
duration.  The  exact  method  is  followed  by  tabulating  the
precise  fractions  of  a  year  of  exposure  over  integral  years  (in
practice  usually  to  the  nearest  month).  The  nearest  duration
is  found  by  takiag  the  nearest  integral  number  of  years,  with
proper  adjustment  so  that  in  cases  where  the  fraction  is  |  the
number  of  cases  counted  as  the  next  higher  integral  year  will
balance  those  counted  as  the  next  lower.  The  mean  duration  is
found  by  subtracting  the  year  of  entry  from  the  year  of  exit  on
the  supposition  that  the  overstated  durations  will  approximately
balance  those  understated.  The  exact  method  is  usually
laborious  and  one  of  the  other  methods  is  therefore  to  be  pre-
ferred if  it  can  be  found  to  give  approximately  correct  results.
The  nearest  duration  method  may  in  some  instances  materi-
ally understate  the  exposures;  for  example,  in  the  first
policy  year  where  the  lapses  at  the  end  of  three  months  may  over-
balance the  lapses  at  the  end  of  nine  months  and  thus  affect  the
exposures  for  the  first  policy  year.     The  mean  duration  method  is

FROM    THE   RECORDS   OF   INSURED   LIVES.  16

based  on  the  assumption  of  an  even  distribution  of  business
throughout  the  calendar  year,  which  may  or  may  not  be  suffi-
ciently in  accordance  with  the  facts  in  any  particular  investiga-
tion. In  general  it  may  be  remarked  that  neither  the  nearest
nor  the  mean  duration  method  should  be  adopted  without  taking
into  consideration  all  the  peculiarities  of  the  fundamental  data.

Those  cases  which  pass  out  of  observation  through  maturity
seldom  introduce  fractional  durations,  but  if  they  arise  fractions
may  be  treated  in  the  same  way  as  for  the  withdrawn.

Deaths  must  always  be  treated  with  accuracy  as  any  error
would  be  of  vast  importance  compared  with  a  similar  error  in
the  exposures.  As  a  life  is  treated  as  exposed  to  risk  during  the
whole  year  of  death  the  duration  would  naturally  be  taken  to
the  policy  anniversary  following  death.  The  curtate  (next
lower  integral)  duration,  however,  may  be  recorded  provided
that,  in  tabulating  the  results,  the  exposures  are  adjusted  to
include  the  year  of  death.

As  the  policy  year  method  tabulates  the  experience  in  the  form
which  is  desired  for  the  construction  of  select  tables  it  is  the
method  which  would  naturally  be  employed  in  important  in-
vestigations of  that  character.  It  was  used  as  the  basis  of  the
British  Offices'  Experience  (1863-1893),  the  Medico- Actuarial
Investigation  (1885-1909),  and  the  American-Canadian  Mor-
tality Investigation  (1900-1915).  A  full  description  of  these
applications  of  the  method  may  be  found  in  the  published
volumes  of  those  experiences.

B.  Calendar  Year  Method.

The  calendar  year  method  is  an  important  historical  method,
used  first  by  Mr.  Woolhouse  in  the  17  Offices'  Experience,  and
used  later  in  the  Institute  (20  Offices')  Experience  and  the  Thirty
American  Offices'  Experience.  It  should  be  noted  that  this
method  can  be  used  where  the  policy  details  are  not  sufficiently
complete  for  a  policy  year  method.  In  connection  with  its  use
in  the  Institute  Experience  it  was  stated  that  much  valuable
data  could  not  have  been  included  in  the  experience  had  precise
dates  of  birth,  entry  and  exit  been  asked  for.  All  that  was
available  in  many  cases  was  the  office  entry  age  next  birthday
and  the  calendar  years  of  entry  and  exit.  Policy  year  tabulation
was  therefore  impossible.

16  CONSTRUCTION   OF   MORTALITY   TABLES

The  method  has  however  lost  favor  as  the  basis  of  important
mortality  tables  constructed  from  life  insurance  experience  be-
cause it  does  not  produce  data  in  convenient  form  for  select
mortality  tables.  The  material  may  be  separated  by  calendar
years  after  entry  and  select  tables  compiled  by  an  approximate
method  as  hereafter  described;  but  year  0 — the  calendar  year
of  entry — represents  only  approximately  the  first  six  months
experience  and  year  1  extends  approximately  from  duration  ^
to  1|,  etc.  Therefore,  if  reliable  select  tables  are  desired  the
policy  year  method  should  be  used.  It  should  also  be  noted
that,  under  the  assumption  that  those  entering  and  withdrawing
do  so  in  the  middle  of  the  year,  persons  withdrawing  in  the
calendar  year  of  entry  are  excluded  from  the  experience,  thus
tending  to  understate  the  exposures.

Under  the  calendar  year  method  of  analyzing  exposures  and
deaths  the  calculations  in  connection  with  the  exposures  are
usually  based  upon  the  assumption  of  an  even  distribution  of
issues  throughout  the  year.  Thus  new  entrants  may  be  assumed
to  be  exposed  on  the  average  for  six  months  in  the  year  in  which
the  policies  are  issued  and  similarly  withdrawals  can  be  con-
sidered as  taking  place  in  the  middle  of  the  year  of  exit.

The  new  entrant  is  assumed  to  enter  in  the  middle  of  the
calendar  year,  which  is  the  period  used  as  the  basis  for  grouping
the  exposures.  In  this  first  calendar  year  therefore  he  will  be
treated  as  exposed  to  risk  for  one  half  a  year  at  his  age  at  the
beginning  of  such  calendar  year.  He  will  be  counted  as  exposed
for  the  next  calendar  year  at  his  age  at  the  beginning  of  that
year.  Therefore,  if  the  age  at  date  of  entry  were  calculated  as
the  age  nearest  birthday  or  mean  age  at  that  date,  he  would
enter  as  a  fractional  exposure  for  that  year  at  an  age  5  year  less
and  would  be  exposed  throughout  the  next  year  at  an  age  |  year
more.  So  if  the  age  at  date  of  entry  were  calculated  as  an  in-
teger, X,  the  calendar  year  exposures  would  be  at  the  fractional
ages  X  —  ^,  X  -\-  i,  X  -{-  3/2,  etc.  The  deaths  would  also  have
to  be  tabulated  by  half  ages.  The  completed  age  at  death  would
be  the  half  age  on  the  January  1st  preceding  death.  We  may
find  the  completed  age  by  adding  the  proper  duration  to  the  age
at  entry.  If  death  occurs  in  the  first  calendar  year  following  the
year  of  entry  we  should  add  ^  to  that  age,  if  death  occurs  during
the  second  year  after  the  year  of  entry  we  should  add  I-2-,  etc.
Thus  the  general  rule  for  determining  the  completed  age  would

FROM  THE  RECORDS  OF  INSURED  LIVES.  17

be  to  add  to  the  age  at  entry  (nearest  birthday  or  mean  age)
the  mean  duration  minus  one-half.  For  example,  if  an  insured
were  35  at  the  date  of  issue  in  1890  and  died  in  1912,  his  age  on
January  1,  1912  would  be  35+  (1912  -  1890)  -  h,  or  56^  In
this  way  the  exposures  and  deaths  would  be  grouped  by  half
ages  and  from  the  values  of  qx+i  so  obtained  the  values  of  q^
might  be  interpolated.

It  would  seem  preferable  in  many  cases  to  derive  the  rates  of
mortality  directly  for  integral  ages,  but  in  order  to  accomplish
this  we  must  follow  a  method  which  will  assume  that  integral
ages  among  the  exposed  to  risk  coincide  with  January  1st.
Where  the  practice  of  insuring  lives  at  the  age  next  birthday
exists,  it  is  convenient  to  consider  that  such  ages  will  be  attained
on  the  January  1st  following  entry.  This  is  equivalent  to  the
assumption  that  the  age  next  birthday  is  on  the  average  |  year
greater  than  the  exact  age  at  date  of  entry  and  that  on  the
average  six  months  elapses  between  the  date  of  entry  and  the  be-
ginning of  the  next  calendar  year.  Each  life  will  then  be  exposed
to  risk  for  ^  year  in  the  calendar  year  of  entry  at  an  integral
age  one  year  less  than  the  age  next  birthday  at  issue,  and  the
succeeding  years  of  exposure  will  obviously  be  at  integral  ages
also.  Withdrawals  would  be  exposed  for  |  year  at  the  integral
age  corresponding  to  January  1st  of  the  calendar  year  of  with-
drawal.*

Another  method  that  will  group  the  data  by  integral  ages
consists  of  using  as  the  age  on  January  1st  following  issue  the
age  nearest  birthday  on  January  1st.  If  the  age  so  calculated
is  X,  the  life  would  be  exposed  for  half  a  year  at  age  x  —  1  in  the
calendar  year  of  entry.

If  the  exposures  were  to  be  tabulated  for  integral  ages  by
calculating  the  age  nearest  birthday  on  January  1st  following
the  date  of  entry  or  by  using  as  the  age  on  such  January  1st  the
age  next  birthday  at  entry,  then  the  age  on  January  1st  of  the
year  of  death  may  be  taken  by  adding  to  such  an  age  the  mean
duration  minus  one.  This  gives  precisely  the  same  result  in
the  former  case  as  would  be  obtained  by  calculating  the  age

*  See  Mr.  Geo.  King's  remarks,  J.  I.  A.,  Vol.  XXVII  (pg.  218)  for  alternate
method  of  treating  new  entrants  and  withdrawals.  It  should  be  remembered
that  the  fractional  exposures  in  the  years  of  entry  and  withdrawal  might  be
calculated  accxirately  but  such  refinement  would  destroy  the  convenience  of
the  method.

18  CONSTRUCTION   OF   MORTALITY   TABLES

nearest  birthday  at  the  beginning  of  the  year  of  death.  In
the  latter  case  the  deaths  would  ordinarily  be  given  somewhat
more  accurately  by  calculating  the  age  nearest  birthday  on  the
January  1st  preceding  death  instead  of  employing  the  rule
just  given,  which  uses  the  age  next  birthday  at  issue.  It  must
be  remembered  that  whatever  method  of  determining  the  age
is  used,  each  death  must  be  included  as  an  exposure  for  the
entire  calendar  year  of  death.

For  the  withdrawn,  the  age  at  the  beginning  of  the  year  of
withdrawal  may  be  determined  in  accordance  with  the  con-
siderations just  mentioned.  If  the  age  next  birthday  at  issue
was  used,  however,  it  would  be  satisfactory  to  add  the  mean
duration  minus  one  rather  than  to  calculate  the  age  nearest
birthday  on  January  1st  of  the  year  of  withdrawal  as  is  some-
times done  for  deaths.

C.  Life  Year  Method.

The  third  way  of  analyzing  the  exposures  and  deaths  is  the
life  year  method.  This  might  be  called  the  exact  method  as  it
traces  the  data  from  birthday  to  birthday.  The  deaths  are
grouped  according  to  the  exact  age  last  birthday  at  the  date  of
death.  The  exact  ages  (to  nearest  month,  or  to  one  decimal
place)  at  entry  and  exit  of  all  lives  entering  into  the  experience
may  be  calculated,  each  life  being  ordinarily  exposed  for  a  frac-
tion of  a  year  of  age  at  the  beginning  and  end  of  its  exposure.
Instead,  however,  of  calculating  each  of  these  fractions  with
exactness  an  average  fraction  may  be  assumed,  after  a  sufficient
test,  to  apply  to  all  cases  except  deaths,  which  are  treated  as
exposed  to  risk  for  the  whole  year  of  death.

In  experiences  where  it  is  justified  a  rougher  approximation
may  be  adopted  for  computing  the  exposed.  This  consists  in
taking  the  ages  at  entry  and  withdrawal  as  the  mean  ages  found
by  subtracting  the  calendar  year  of  birth  from  the  calendar
year  of  entry  and  exit  respectively.  This  avoids  fractional
durations  and  is  based  on  the  assumption  that  the  resulting
over-exposures  and  under-exposures  will  balance.  There  is  a
difficulty  presented  if  the  experience  starts  at  a  certain  date
with  a  number  of  lives  then  under  observation  or  closes  with  a
number  existing.  For  such  cases  the  ages  at  entrance  and  exit
may  be  taken  as  the  age  nearest  birthday  or  determined  by  some
other  suitable  method.

FROM   THE   RECORDS   OF   INSURED   LIVES.  19

Another  method  of  estimating  the  exposures  is  available  by-
using  the  ages  nearest  birthday  at  entry  and  exit  for  the  entire
data  except  deaths.  Under  this  method  the  error  involved  in
the  age  at  entry  or  exit  for  any  one  life  cannot  exceed  six
months,  while  the  mean  age  may  be  incorrect  by  almost  a  year;
but  under  both  methods  the  largest  possible  error  in  duration  is
approximately  one  year.

The  life  year  method,  while  making  possible  the  strictest
accuracy  if  the  ages  are  exactly  calculated,  has  a  very  narrow
scope  of  usefulness  in  life  insurance  experience,  because  it  does
not  produce  the  results  in  the  form  of  select  tables  and,  if  applied
with  exactness,  is  laborious.  It  has  been  employed  in  tabulating
mortality  among  annuitants.*

A  general  warning  should  be  given  regarding  the  use  of  ap-
proximations which  have  been  frequently  suggested  in  connection
with  all  these  methods.  In  any  important  investigation  the
effects  of  such  approximations  should  be  tested  before  they  are
adopted.  For  example,  where  an  office  accepts  applicants  at
rates  for  the  nearest  birthday  there  will  be  a  tendency  for  persons
to  insure  just  before  the  nearest  birthday  changes.  In  such
case  the  nearest  birthday  at  entry  would  on  the  average  some-
what understate  the  true  age.f  If  lives  are  accepted  at  rates
for  the  age  next  birthday  the  tendency  will  be  to  insure  more
often  just  before  than  just  after  a  birthday,  and  the  result  of
using  the  nearest  age  may  be  to  overstate  the  age  slightly.
Again  a  company's  business  may  not  be  evenly  distributed  over
the  year,  because  of  pressure  at  the  end  of  its  financial  year  or
toward  the  culmination  of  agency  contests,  and  methods  de-
pending upon  such  even  distribution  may  produce  too  great  an
error.  Considerations  such  as  these  should  be  taken  into  account
in  determining  the  methods  to  use.

Mortality  investigations  frequently  cover  the  experience  under
all  policies  in  force  within  a  specified  period.  It  then  happens
that  many  lives  will  enter  the  experience  at  the  beginning  of
the  period  after  having  been  insured  for  some  time.

*  For  applications  of  the  method  see  J.  I.  A.,  Vol.  XXXI,  pages  171-5
and  179-181.

t  If  a  mortality  experience  is  to  be  used  as  a  basis  for  premium  rates  charged
at  nearest  ages  there  is  no  objection  to  such  understatement  of  ages  in  the
data  with  resulting  slight  increase  in  mortality  rates;  for  the  companies  will
thus  have  provision  in  the  mortaUty  table  for  the  effect  of  charging  premiums
at  nearest  age.

20  CONSTRUCTION   OF   MORTALITY   TABLES

If  the  policy  year  method  is  to  be  used  the  experience  is  gen-
erally taken  between  the  policy  anniversaries  in  the  first  and
last  years  of  the  period.  By  this  means  fractional  exposures  are
avoided  at  the  beginning  in  the  case  of  lives  brought  forward.
Care  must  be  taken  however  to  see  that  the  policy  years  elapsed
since  original  entry  are  properly  recorded  for  use  in  tabulating
select  tables.  If  other  methods  are  used  the  manner  of  handling
the  lives  that  enter  the  experience  in  this  way  must  be  consistent
with  that  employed  for  lives  first  insured  during  the  period  of
investigation.

It  is  customary  to  allow  policyholders  a  grace  of  thirty  days  or
one  month  for  the  payment  of  renewal  premiums.  If  death
occurs  during  the  grace  period  while  the  premium  is  unpaid  the
claim  is  allowed,  the  overdue  premium  being  deducted  from  the
face  of  the  policy..  As  such  deaths  will  be  recorded  in  a  mortality
experience  theoretical  accuracy  might  seem  to  require  treating
every  lapsed  policy  as  exposed  to  risk  for  one  month  after  the
due  date  of  the  premium,  in  order  that  the  corresponding  ex-
posures may  be  recorded;  or,  if  these  fractional  exposures  are
not  taken  into  account,  then  it  might  appear  that  no  account
should  be  taken  of  the  corresponding  deaths.

It  is  impossible,  however,  to  ascertain  what  deaths  should
properly  be  eliminated  under  the  latter  alternative  because  we
ought  to  exclude  only  those  cases  where  the  premiums  would
not  have  been  paid  had  the  person  insured  survived  the  grace
period.  Furthermore  there  are  practical  objections  to  treating
the  grace  period  on  lapsed  policies  as  exposures,  because  com-
panies will  continue  to  grant  a  grace  privilege  and  their  premiums
must  be  suflficient  to  provide  that  benefit.  By  including  the
deaths  and  excluding  these  fractional  exposures  the  mortality
table  will  give  premiums  which  contain  a  provision  for  this.
It  is  therefore  the  usual  practice  to  include  the  deaths  and  ex-
clude the  exposures.*

When  a  policy  lapses  after  having  been  carried  a  short  time,
it  is  customary  to  grant  an  automatic  nonforfeiture  feature.
If  automatic  extended  insurance  is  given  then  it  must  be  de-
cided whether  the  exposure  is  to  cover  the  period  of  risk  under
the  original  form  of  policy  only,  or  whether  the  risk  under  the

*  The  British  Offices'  Experience  is  an  important  exception  to  this  rule.
The  nearest  duration  method  was  modified  to  take  account  of  the  grace  period.
See  Appendix  M,  volume  entitled  "Account  of  Principles  and  Methods."

FROM   THE  RECORDS  OF  INSURED  LIVES.  21

term  extension  is  to  be  included.  When  the  decision  is  made  the
deaths  are  taken  to  correspond.  If  the  risk  under  the  extended
term  insurance  is  to  be  included  the  fractional  durations  at  the
expiry  of  the  term  insurance  must  be  treated  in  the  same  way
as  the  fractional  exposures  of  the  withdrawals.

Many  policies  are  changed  at  lapse  to  reduced  paid-up  in-
surance either  automatically  or  upon  request.  If  the  experience
is  based  upon  amounts  insured,  such  policies,  if  included  in  the
experience,  must  be  treated  as  withdrawals  for  the  amount  of
the  reduction  and  the  balance  continued.  This  is  awkward  and
it  may  be  thought  best  to  exclude  all  data  from  the  date  of  lapse.

It  must  also  be  kept  in  mind  in  treating  paid-up  and  extended
insurance  that  the  character  of  the  mortality  under  these  options
may  vary  considerably  from  that  under  policies  in  full  force.
It  may  therefore  be  best  in  some  circumstances  to  make  separate
investigations  for  such  lapsed  policies  to  determine  the  company's
attitude  toward  the  non-forfeiture  features  and  to  exclude  the
data  from  the  company's  general  experience.

The  question  also  arises  whether  to  include  policies  after  the
amount  of  insurance  has  been  changed.  Except  where  the
investigation  is  by  amounts  iasured  a  reduction  in  amount
presents  no  diiO&culties,  but  as  an  increased  insurance  is  usually
granted  only  upon  medical  examination  such  lives  would  be
select  from  the  date  of  change.  If  an  aggregate  table  is  to  be
constructed  this  might  be  disregarded  but  for  a  select  table
recognition  must  be  taken  of  this  fact.  Because  of  such  com-
plications it  will  sometimes  be  best  to  exclude  one  or  both  of  these
classes  of  changed  policies.

The  same  principles  will  control  the  decision  as  to  the  inclusion
of  policies  changed  in  form,  for  changes  to  lower  premium  forms

If  a  policy  lapses  and  is  afterwards  reinstated  upon  evidence  of
good  health  it  must  be  decided  whether  the  exposure  should
terminate  at  the  date  of  lapse  and  start  again  at  the  date  of
reinstatement,  or  whether  the  temporary  lapse  should  be  dis-
regarded. Where  the  period  between  the  date  of  lapse  and  date
of  reinstatement  is  short  and  only  a  personal  certificate  of  health
has  been  furnished,  no  appreciable  error  will  be  introduced  by
ignoring  the  lapse  and  reinstatement.  Where  considerable  time
elapses  over  which  the  policy  is  not  exposed  to  risk  and  a  new
medical  examination  is  made,  it  may  be  thought  better  to  take

22  CONSTRUCTION   OF  MORTALITY  TABLES.

account  of  the  lapse  and  terminate  the  exposure  at  that  point.
The  subsequent  reinstatement  may  then  be  treated  as  a  new
policy;  but,  bearing  in  mind  that  a  company  is  not  always  so
particular  in  reinstating  a  lapsed  policy  as  in  issuing  a  new  one,
and  often  reinstates  an  impaired  life  covered  by  extended  in-
surance because  the  mortality  loss  will  be  less  under  the  premium-
paying  policy,  it  may  be  thought  better  to  eliminate  entirely
from  the  investigation  the  exposures  under  such  reinstated
policies.  It  is  not  likely  that  any  material  effect  upon  the  results
of  the  investigation  would  be  caused  either  by  including  or  ex-
cluding these  exposures.

In  the  construction  of  mortality  tables  many  points  will  arise
which  call  for  the  independent  judgment  of  the  compiler,  and
when  the  tables  are  published  it  is  usual  to  state  how  these
different  points  have  been  handled.

When  constructing  such  tables  it  is  sometimes  desired  to
investigate  collateral  questions  such  as  the  relative  mortality
by  plan  of  insurance,  amount  of  insurance,  residence,  occupation
or  other  features  of  the  risk.  It  is  therefore  necessary  to  plan
the  card  on  which  the  policy  details  are  to  be  recorded  so  that
provision  will  be  made  for  the  information  necessary  for  such
investigations.

Chapter  IV.     Handling  the  Data.

The  compilation  of  a  mortality  experience  is  usually  carried
out,  as  has  been  mentioned,  by  recording  the  data  on  cards
because  of  the  ease  of  sorting  the  material  by  age  at  issue,  dura-
tion, mode  of  exit  or  any  other  feature.  It  is  necessary  to
exercise  care  to  see  that  the  sources  from  which  these  details
are  drawn  are  reliable,  particularly  in  regard  to  the  mode  of
exit;  for  it  is  essential  that  the  deaths  be  recorded  correctly.
It  may  be  worth  while  to  check  all  cases  recorded  as  dead  with
the  claim  records,  if  they  are  available.  If  the  cause  of  death
is  to  be  used  for  a  subsequent  investigation  of  the  relative
frequency  of  the  different  causes,  such  a  check  may  be  obtained
with  little  or  no  extra  labor.

If  the  table  to  be  constructed  is  an  aggregate  table  or  simply
excludes  certain  early  years  of  duration,  as  in  the  case  of  a
truncated  table,  it  is  necessary  to  record  the  age  at  entrance
into  the  experience  and  at  exit.  Equivalent  facts  would  be
obtained  by  recording  the  age  at  entry  and  duration,  from  which
the  age  at  exit  can  be  obtained  by  addition.  When  select  tables
are  desired  it  is  necessary  to  show  the  duration  instead  of  the
age  at  exit.

In  writing  the  cards  the  policy  records  must  be  examined  care-
fully to  exclude  the  data  which  is  not  desired,  as  for  example,
under-average  lives,  female  lives,  policies  continued  under  non-
forfeiture provisions,  residents  outside  the  temperate  zone,  etc.

If  the  mortality  table  is  to  be  based  on  lives  insured,  the  date  of
birth  and  name,  or  all  initials,  should  be  entered  so  that  dupli-
cates may  be  eliminated  as  described  in  Chapter  III.  For  a
table  based  on  amounts  insured  care  must  be  taken  to  record
properly  cases  in  which  a  change  in  the  amount  of  insurance  has
occurred.  If  such  cases  are  to  be  included  in  the  experience
after  the  date  of  change  it  will  probably  be  best  to  write  two
cards,  one  for  the  lesser  amount  and  one  for  the  balance,  with
proper  dates  of  entrance  and  withdrawal  to  make  them  equivalent
to  the  case  as  a  whole.  When  an  experience  both  by  policies
and  by  amounts  is  desired,  the  lesser  amount  may  be  counted  as
one  policy  while  the  balance  may  be  indicated  as  "no  policy."

23

24  CONSTRUCTION   OF  MORTALITY   TABLES

Then  in  combination  they  will  count  as  one  policy  for  the  full
amount.

For  select  tables  cases  re-examined  medically,  such  as  in-
creased amounts,  changes  to  lower  premium  forms,  and  possibly
reinstatements,  if  included  in  the  investigation,  would  usually
be  recorded  as  withdrawn  at  the  time  of  change  or  lapse,  and
additional  cards  would  be  written  to  tabulate  them  as  new
entrants  after  such  dates.

Policies  which  enter  the  experience  at  the  beginning  of  the
period  covered  by  the  investigation,  having  been  previously
issued,  must  be  specially  treated.  If  only  an  aggregate  table  is
desired  the  age  at  entry  may  be  recorded  as  the  age  at  date  of
entering  the  period.  If,  however,  select  tables  are  to  be  con-
structed the  card  must  show  the  age  at  issue,  duration  at  entry
into  the  experience,  and  duration  at  exit.

Those  policies  which  are  still  in  effect  at  the  close  of  the  period
of  investigation  will  have  the  duration  at  that  time  recorded.

The  form  of  card  to  be  used  should  be  selected  with  care  to
include  all  information  necessary  to  permit  of  the  easiest  and
most  accurate  recording,  and  to  make  the  elimination  of  dupli-
cates, sorting  and  tabulating  as  rapid  as  possible.  The  volume
entitled  "Account  of  Principles  and  Methods"  of  the  British
OflSces'  Experience  (1863-1893)  gives  an  excellent  example  of
the  points  which  must  be  considered.  The  card  shown  on  the
following  page  is  one  of  several  forms  used  in  that  experience.

The  letters  S.A.  indicate  that  this  card  was  to  be  included  in
the  data  both  for  the  select  and  also  for  the  aggregate  tables.
The  age  at  entry  was  taken  as  the  age  nearest  birthday.  There-
fore +  was  marked  after  the  year  of  birth  in  this  individual  case
to  indicate  that  the  age  at  entry  was  the  difference  between  the
year  of  entry  and  one  more  than  the  year  of  birth.  The  data
were  analyzed  by  policy  years,  and  the  duration  recorded  on  the
cards  for  deaths  was  the  curtate  duration,  or  the  duration  at
the  commencement  of  the  policy  year  of  death.  Cards  of
different  color  were  used  for  male  and  female  lives.

It  is  interesting  to  note  that  a  preliminary  test  showed  that  a
general  alphabetical  arrangement  of  the  cards  would  not  satis-
factorily eliminate  duplicates.  The  cards  were  therefore  sorted
chronologically  by  date  of  birth.  After  noting  the  duplicates
detected  in  this  way,  the  cards  in  each  year  of  birth  were  sorted
alphabetically  and  again  examined  for  duplicates.

PROM  THE   RECORDS   OF  INSURED   LIVES.

25

S.A.
NO 2356

Old  Policies.                        N^^
£....100.

CLASS....0

PROFIT  OR  NOT....P

(Latimer    ...

LIFE  \

\  Darsie

DATE—

OF  BIRTH

D.

7

M.          Year.
8. 1826 +

OF  ENTRY...

1

1 1850

In  1863

1863.

OF  EXIT

9q

2               1888

Duration  before  1863

13.

Duration  of  Policy.

38......

Age  at  Entry

Age  in  1863

23

Age  at  Exit

MODE  OF  EXIT.

(D)

REMARKS.

J.                                                                                              ^^

(Select  and  Aggregate  Tables.)

In  compiling  the  census  returns  of  1890,  the  United  States
Census  Office  made  the  first  use  of  the  system  of  punched  cards,
which  was  devised  by  Dr.  Hollerith  and  permits  of  mechanical
sorting  and  tabulating.  This  system  makes  use  of  a  card
printed  with  a  series  of  columns  running  from  0  to  9.  If  the
card  is  to  represent  a  policy  issued  at  age  35,  the  "3"  in  the
tens  column  and  the  "5"  in  the  units  column  representing  the
age  will  be  punched  out.     By  means  of  code  numbers  the  occupa-

26  CONSTRUCTION   OF  MORTALITY  TABLES

tion  or  other  information  may  be  entered  upon  the  Hollerith
cards;  for  example,  two  columns  will  cover  100  and  three  columns
1,000  different  occupations.  By  an  ingenious  mechanical  device
the  capacity  of  the  cards  may  be  somewhat  increased  beyond  the
100  or  the  1,000  without  increasing  the  number  of  the  columns
beyond  the  two  or  three.  After  the  cards  are  punched  they
are  sorted  mechanically  at  a  high  rate  of  speed,  it  being  claimed
that  15,000  cards  may  be  sorted  in  one  hour.  Another  machine
specially  designed  will  add  four  items  on  the  cards  at  a  speed
slightly  lower  than  that  of  the  sorting  machine.  Machines  may
be  built  on  the  same  lines  to  add  different  numbers  of  items.
The  student  is  referred  to  two  articles  (T.  A.  S.  A.,X1,  252,  276)
for  further  information  concerning  the  Hollerith  system.

Another  system  which  employs  similar  punched  cards  is  that
put  out  by  the  Powers  Accounting  Machine  Company.  These
machines  were  used  in  carrying  out  the  American-Canadian
Mortality  Investigation.  A  copy  of  the  card  used  in  that  ex-
perience is  given  on  page  27.     (See  Page  11,  A.  C.  M.  I.)

Volume  1  of  the  investigation  states  "The  machines  used  for
perforating  the  cards  were  so  constructed  that  the  cards  auto-
matically came  into  place  for  perforation.  The  hole  to  be
punched  in  each  column  of  the  cards  is  selected  by  a  sliding
scale  or  bar  and  the  perioration  is  done  by  electricity.  Two
cards  may  be  prepared  at  one  time,  and  accordingly,  cards  of
two  colors  are  inserted  alternately  in  the  package  of  blank  cards
and  placed  in  the  machine.  The  verification  of  the  accuracy
in  perforating  the  cards  may  be  done  in  two  ways,  either  by
comparing  the  perforated  card  with  the  original  record,  or  by
using  a  verifying  machine  so  constructed  that  when  the  operator
strikes  an  incorrect  key  or  when  the  key  struck  does  not  agree
with  the  hole  already  in  the  card  the  machine  automatically
stops  and  thus  attention  is  called  to  an  error  on  the  part  of  the
original  perforating  clerk  or  of  the  operator  of  the  verifying
machine.

"A  great  advantage  of  this  perforating  machine  is  that  in  certain  types  of
perforation  the  keys  need  not  be  changed  if  the  same  figures  appear  in  a
number  of  cards.  If,  for  example,  the  cards  were  all  issues  of  the  year  1914
and  were  issued  in  the  State  of  New  York  on  the  twenty  year  endowment
insurance  plan,  then  in  the  proper  fields  the  figiires  14,  3  and  38  would  remain
automatically  in  place  until  all  the  cards  coming  within  that  group  had  been
perforated.  Another  advantage  is  that  the  complete  setting  of  the  machine
for  perforation  can  be  read  along  the  edge  of  the  machine  before  touching  the
lever  which  perforates  the  card."

FROM   THE    RECORDS   OF   INSURED   LIVES.

27

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1-H

28  CONSTRUCTION    OF   MORTALITY   TABLES

The  following  statements  regarding  the  sorting  and  tabulating
machines  will  give  a  good  idea  of  the  remaining  operations  of
this  system.

"The  sorting  machine  if  kept  suppUed  with  cards,  will  classify  12,000  per
hour.  This,  of  course,  refers  to  sorting  any  one  column.  If  the  cards  are
to  be  sorted  by  age,  two  sortings  are  necessary, — first,  into  groups  of  decennial
ages,  i.e.,  20-29,  30-39,  etc.;  and  then  into  separate  age  groups.  Advantages
of  this  type  of  machine  are  that  the  cards  are  placed  by  machine  in  compart-
ments which  are  easy  to  empty,  as  they  do  not  involve  stooping,  and  that
all  cards  of  a  specified  number  can  be  segregated,  leaving  the  cards  with  other
nimabers  in  the  same  numerical  or  other  order.  Several  machines  used  by
the  bureau  had  counters,  technically  known  as  coimting  sorters,  attached  to
the  sorting  machines.  These  machines  count  the  number  of  cards  going  into
each  compartment  or  pocket  and  give  also  the  total  number  of  cards  in  all
the  pockets.

"This  machine  (the  tabulator)  prints  on  strips  of  paper  such  fields  in  the
cards  as  are  desired.  This  is  known  as  selecting  mechanically  the  desired
coliunns  of  the  cards  to  be  recorded  on  the  list.  It  can  be  so  arranged  as  to
record  letters  instead  of  the  figures  appearing  on  the  card.  For  example,  the
code  number  for  the  termination  of  the  policy  by  death  is  9,  but  it  appears  on
the  list  prepared  by  the  machine  as  '  D.'  From  one  to  seventeen  columns  may
be  added.  The  work  of  this  machine  can  best  be  imderstood  by  means  of
the  following  example:

Example  of  Work  bt  Tabttlator-Printer.

Explanation  of  Columns  of  Figures  or  Symbols.
(o)  Year  of  issue;  i.e.,  2  stands  for  1902,  12  for  1912  and  8  for  1908.
(6)  Age  at  entry.

(c)  Duration  in  years,

(d)  Sex  (M  =  Men).

(e)  Mode  of  termination  (L  =  Lapsed).
(/)  Code  number  for  plan  of  insurance,

(<g)  Code  number  for  habitat  at  date  of  application  for  insurance.

(h)  Nmnber  of  policies.

0)  Amoimt  of  insurance  to  nearest  \$100;  e.g.,  10  =  \$1,000,  and  25  =  \$2,500

(a)  (6)

(c)

(d)

(e)

(/)

(9)

(h)

U)

2  20

M

L

1

16

30

12  20

M

L

1

16

10

12  20

M

L

2

16

150

13  20

M

L

1

16

50

11  20

M

L

3

16

100

8  20

M

L

4

16

20

10  20

M

L

1

16

10

10  20

M

L

1

16

30

1  20

M

L

1

16

10

6  20

M

L

2

16

10

11  20

M

L

1

16

10

11  20

M

L

3

16

300

(Total  niunber  of  policies)  12  (for  ins.  of)  730

[\$73,000]

FROM  THE  RECORDS  OF  INSURED  LIVES.  29

"The  parts  in  brackets  are  not  printed  by  the  machine  but  are  added  for
the  sake  of  clearness.

"After  the  cards  have  been  sorted  into  the  desired  groups,  special  cards
known  as  'total'  cards  were  placed  between  these  groups.  The  machine  con-
tinues to  print  as  shown  above  until  it  comes  to  a  'total'  card  when  the  desired
summations  are  made;  the  machine  then  commences  to  tabulate  the  ensuing
group.  After  all  the  cards  in  the  machine  have  been  hsted  and  added,  the
machine  automatically  stops.  An  operator  can  handle  two  or  more  machines,
as  it  is  necessary  merely  to  keep  them  supplied  with  cards.  Cards  pass
through  the  machines  at  the  rate  of  2,500  per  hour  when  there  are  large  groups
and  about  2,000  per  hour  when  there  are  small  groups  necessitating  frequent
summation.

"  A  study  of  the  foregoing  example  of  listing  will  indicate  that  the  Tabulator-
Printer  is  of  great  value  in  detecting  errors.  In  the  example  given,  the  insured
were  all  men  of  twenty  years  of  age  who  had  lapsed  their  poUcies  with  a  diu-a-
tion  of  one  year  and  were  residing  in  State  No.  16  (Illinois).  Any  errors  re-
sulting from  putting  cards  wrongly  in  any  of  these  categories  would  be  easily
detected.  The  same  would  hold  true  in  other  sortings,  such  as  those  among
women  who  had  died  in  the  third  insurance  year.  In  that  case  column  (c)
should  be  '  3 ' ;  column  (d)  should  be  '  F ' ;  and  column  (e)  shovild  be  '  D.'  Again,
if  the  group  consisted  of  men  aged  25  whose  poUcies  were  existing  at  the  close
of  the  investigation  and  who  had  taken  their  poUcies  in  1905  on  the  twenty
payment  life  plan  when  they  were  resident  in  Massachusetts,  column  (a)
should  be  '5';  column  (6)  should  consist  only  of  '25';  column  (d)  of  'M';
column  (e)  of  'E';  column  (/)  of  '2';  column  (g)  of  '25';  column  (c)  should
be  blank."

From  the  above  descriptions  of  the  mechanical  systems  in
use  it  will  be  seen  that  their  invention  has  made  it  possible  to
carry  out  expeditiously  large  mortality  investigations  which
would  involve  extraordinary  effort  if  carried  out  by  hand.  The
successful  operation  of  these  systems,  however,  requires  careful
planning  of  every  detail  that  is  to  be  recorded.

In  connection  with  these  mechanical  systems  it  should  be
kept  in  mind  that  in  most  cases  it  would  be  impracticable  to
punch  the  cards  directly  from  the  company's  records.  It  is
customary,  therefore,  to  prepare  preliminary  sheets  on  which
can  be  written  the  data  necessary  for  the  cards.  The  duration
or  age  at  exit  may  be  calculated  from  the  original  records  by
the  person  making  out  the  working  sheet,  thus  reducing  the
punching  of  the  cards  to  a  simple  and  rapid  operation.  The
form  of  sheet  used  in  the  American-Canadian  Mortality  In-
vestigation is  shown  on  the  following  page.

This  form  should  be  compared  with  the  specimen  card  which
is  shown  on  page  27  and  which  was  punched  from  it.

30

CONSTRUCTION   OF   MORTALITY  TABLES

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FROM   THE   RECORDS   OF  INSURED    LIVES.  31

In  investigations  carried  out  by  means  of  written  cards,  no
preliminary  working  sheets  would  ordinarily  be  necessary,  as
the  cards  could  be  written  directly  from  the  original  records.
It  should  be  noted  how  the  card  on  page  25,  used  in  the  British
Offices'  Experience,  is  made  up  conveniently  for  copying  from
the  original  records  and  then  for  computing  the  durations  and
ages.

Before  tabulating  the  cards,  whether  they  be  on  the  written  or
punched  form,  it  is  desirable  to  have  some  rough  check  on  their
accuracy  even  though  they  have  already  been  checked  individu-
ally with  the  source  from  which  the  data  were  drawn.  This  is
particularly  advisable  if  the  records  of  several  companies  are
being  combined.  In  the  American-Canadian  Mortality  In-
vestigation it  is  stated :

"It  seemed  essential  to  have  some  check  on  the  data  so  as  to  determine
whether  a  section  of  the  experience  relating  to  existing,  lapsed  or  deaths  had
been  omitted.  So  far  as  concerns  the  principal  companies  which  furnished
their  data  on  perforated  cards,  very  careful  checks  were  made  by  these  com-
panies, including  calculation  of  expected  deaths  by  a  standard  table  and
comparison  with  the  actual  death  losses.  For  the  other  companies  the  bureau
tabulated  the  existing,  discontinued  and  dead,  according  to  age,  and  ascer-
tained if  they  were  reasonable.  The  death  losses  were  also  subdivided  by
insurance  years  and  comparison  made  between  the  groups  by  ages  and  in-
surance years.     This  would  probably  have  brought  out  any  material  error.

"All  cases  were  investigated  which  showed  that  the  age  of  entry  was  over
70  as  the  niunber  of  such  might  be  few  and  therefore  errors  in  perforating
might  affect  the  results.  Where  the  age  at  death  was  over  100  the  company
reporting  the  case  was  asked  to  verify  the  original  record."

Chapter  V.     Tabulating  the  Data.

Before  sorting  the  cards  on  which  the  details  have  been  re-
corded, careful  plans  should  be  made  for  obtaining  the  exposed
to  risk  in  the  easiest  manner  possible.  A  continuous  process  is
used  by  which  Ex+i,  the  exposed  to  risk  at  age  x  -\-  1,  is  derived
from  Ex.  The  formula  to  be  employed  will  depend  upon  the
method  of  analyzing  the  exposures  and  deaths — policy,  calendar,
or  life  year  method — and  upon  the  character  of  the  mortality
table  to  be  constructed.*

A.  Policy  Year  Method.

Let  us  first  consider  the  construction  of  an  aggregate  table  by
the  policy  year  method.

Let  rix  =  New  entrants  at  age  x.

ex  =  Existing  at  age  x  at  close  of  observations.

ffx  =  Those  under  observation  at  age  x  when  the  observa-
tions began.

Wx  =  Withdrawals  at  completed  age  x.

Qx  =  Aggregate  of  fractional  durations  beyond  com-
pleted age  X  arising  from  the  Wx  withdrawals.

dx  =  Deaths  occurring  between  ages  x  and  x  +  I.

Let  it  be  assumed  that  Wx  includes  both  the  withdrawals  and
the  matured.  If  it  is  desirable  to  have  them  separated  for
other  purposes,  separate  symbols  may  be  assigned  to  each,  but
the  formulae  will  hold  with  the  corresponding  substitutions.
It  will  be  noticed  that  the  completed  age  at  death  is  used,  i.e.,
the  age  at  issue  plus  the  curtate  duration.  Therefore  the  dx
deaths  must  be  treated  as  exposed  to  risk  to  age  x  +  1.

Then

XXX  X  X— 1

Ex  =   E^x  +  Z<^x  -   T.Wx  +  Qx  -    E«x  -   E^x
0  0  U  U  0

and

Ex+i  =  Ex  -\-  rix+i  +  Cx+i  —  'M'x+i  +  (^x+i  —  ^i)  —  ex+i  —  Ox.    (1)

*  The  authors  wish  to  acknowledge  their  indebtedness  to  Mr.  Robert
Henderson  for  his  concise  explanation  of  formulae  and  methods  of  treating
ages  and  diu'ations  contained  in  his  monograph  "Mortality  Laws  and  Statis-
tics" published  by  John  Wiley  &  Sons,  Inc.

32

FROM   THE   RECORDS   OF   INSURED   LIVES.

33

It  will  be  seen  that  the  use  of  gx  assumes  that  the  exact  dura-
tion method  has  been  used.  If  nearest  or  mean  durations  are
employed  the  withdrawals  take  place  at  integral  ages  and  Qx
becomes  zero  for  all  values  of  x.

In  order  to  carry  out  this  formula,  the  information  may  be
recorded  on  one  sheet  similar  to  the  following.

v

^

(1)
New
En-
trants.

(2)
Enter-
ing
Obser-
vation.

(3)  =

(1)
+  (2).

(4)
With-
draw-
als.

(5,

Exist-
ing.

(6)

(7)  =
(4)  -|-(5)
-F(6)  Pre-
vious line.

(8)

Sx=(3)

-(7) +(8)

Previous

line.

(9)

15

16

17

1

18

19

The  cards  would  be  sorted  by  the  mode  of  entrance  corre-
sponding to  columns  (1)  and  (2),  if  they  were  not  already  so
separated  because  of  their  different  years  of  issue.  Each  set
of  cards  would  then  be  sorted  by  age  at  entrance  into  the  ob-
servations and  the  totals  for  each  age  recorded.  The  next  step
would  be  to  sort  them  all  in  accordance  with  the  mode  of  exit  and
then  by  age  at  exit.  In  tabulating  the  number  of  withdrawals
at  age  x  we  shall  have  the  sum  of  the  fractional  exposures,  gx,
to  tabulate  also  if  the  exact  duration  method  was  used.  If,
however,  we  tabulate  the  expression  {wx  +  gx-i  —  gx)  in  column
(4),  the  value  of  column  (7)  will  be  {wx  +  gx-\  —  gx  -{-  ex  +  6x-i),
which  is  the  quantity  that  enters  negatively  in  equation  (1).

This  method  of  tabulating  might  be  described  as  assuming
that,  of  the  Wx  withdrawals  at  completed  age  x,  we  consider  gx
as  withdrawn  at  age  x  +  1  and  Wx  —  gx  at  age  x.  A  separate
sheet  or  column  might  be  provided  for  recording  the  values  of
gx  and  of  gx-i  —  gx-  It  will  be  noticed  that  this  latter  value  may
be  negative,  in  which  cases  its  absolute  value  is  deducted  from  Wx.

Where  select  tables  are  to  be  constructed  the  exposed  to  risk
must  be  tabulated  for  each  age  at  issue  separately  and  the
formulae  become

Eix]+t  =  nix]  +  T,<^[x]+t  —  Zw[x]+t  +  gix\-\-t

1  0

t  t-\

1  u

34

CONSTRUCTION   OF  MORTALITY  TABLES

and

Eix]+t+i  =  E[x]+t  +  o-[x]+t+i

—  W[x]+t+i  +  (gix]+i+i  —  giz]+t)  —  e[x]+t+i  —  6[x]+t        (2)

where  exact  durations  have  been  used.

If  the  nearest  or  mean  durations  are  taken,  gix]+t  vanishes  and
we  have

E[x]+t+i  =  -£^[i]+t  +  <r[x]+t+i  —  (w[g]+t+i  +  €ix]+t+i  +  d[x]+t)  (3)

In  all  of  the  foregoing  equations  it  has  been  assumed  that
observations  began  and  ended  on  policy  anniversaries  in  the
first  and  last  calendar  years  of  the  investigation,  so  that  there
are  no  fractional  durations  for  a^  and  e,;  otherwise  the  fractional
exposures  must  be  treated  in  a  manner  similar  to  those  arising
from  the  withdrawals.

In  order  to  produce  the  values  of  E[x]+t+i  by  equation  (3)  it
will  be  seen  that  the  sorted  cards  may  be  tabulated  on  sheets
drawn  up  somewhat  as  follows,  using  a  separate  sheet  for  each
age  at  issue.

Age  at  Issue

Number  of  Kntrant«

1

Dura-
tion.

(1)
Entering
Observa-
tions.

(2)

With-
drawals.

(3)
Existing.

(4)

(5)

Decrement

(2)-H(3)-(l)

-h  Previous

Line  (4).

(6)

Exposed  to  Risk

(6)  Previous

Line  —(5).

0

1

2

3

4

The  exposed  to  risk  at  duration  0  would  be  found  by  sub-
tracting the  withdrawals  at  that  duration  from  the  number  of
entrants  shown  at  the  top  of  the  sheet.  There  would  naturally
be  none  entering  the  observations  or  existing  at  duration  0.

To  have  the  cards  in  the  proper  order,  they  must  be  sorted
by  age  at  issue,  then  by  mode  of  exit  and  finally  by  duration  at
exit.  The  cards  for  <Tix]+t  must  also  be  sorted  by  duration  at
entrance  into  the  experience.

FROM   THE   RECORDS   OF  INSURED   LIVES.  35

If  the  exact  duration  method  is  used  for  the  withdrawals  the
cards  should  show  and  be  sorted  by  the  curtate  duration.  For
each  duration  there  must  then  be  calculated  the  value  of  g[x]+t-
This  number  may  be  tabulated  as  withdrawing  at  duration  t  -f  1
and  the  balance  of  the  withdrawals,  (w[x]+t  —  g[x]+t),  may  be
placed  opposite  the  curtate  duration,  t.  This  will  permit
Eix]+t  to  be  computed  in  the  same  way  as  where  mean  or  nearest
durations  were  employed.  In  writing  down  the  withdrawals
on  the  sheet  two  entries  will  be  made  for  each  duration,  except
duration  0,  because  of  the  withdrawals  thrown  forward  and  those
thrown  back.  It  would  therefore  be  well  to  have  the  column
for  withdrawals  subdivided  for  each  duration  by  a  light  horizontal
line.

Having  thus  obtained  the  exposed  to  risk  the  select  mortality
rates  may  be  computed,  for

Ql']+t

E

w+<

The  period  of  selection  may  also  be  determined  by  combining  the
data  on"  the  sheets  for  the  several  ages  at  entry  as  described  in
Chapter  I.  Where  an  aggregate  table  is  wanted  in  addition  to
select  tables,  the  deaths  and  exposed  to  risk  for  each  attained
age  may  be  footed  from  the  sheets  and  the  ratios  of  these  totals
calculated.

It  may  be  desired  to  obtain  an  ultimate  or  a  truncated  table
only.  If  we  let  t  be  the  maximum  duration  excluded  from  such
an  experience  then  every  policy  in  force  not  more  than  t  years
would  be  eliminated  and  every  other  policy  would  be  recorded
as  entering  at  age  x  -\-  t.  In  other  respects  the  same  method
could  be  followed  and  the  same  formulae  used  as  for  an  aggregate
table.  To  construct  a  true  ultimate  table,  however,  it  would
not  be  possible  to  determine  t  beforehand  and  it  would  be  neces-
sary to  list  the  select  data  for  that  purpose.  The  method  just
mentioned  is  therefore  practicable  only  for  a  truncated  table
where  the  experience  to  be  excluded  is  fixed  before  the  investiga-
tion is  tabulated.

In  the  British  Offices'  Assurance  Experience  the  withdrawals,
that  is  lapses  and  surrenders,  were  designated  by  iy[x]+«,  while
the  matured,  called  "Terminations,"  were  treated  separately
under  the  symbol  7'[i]+«.  For  both  classes  the  nearest  duration
method  was  used,  properly  modified  to  include  the  days  of  grace

36  CONSTRUCTION   OF  MORTALITY  TABLES

in  the  exposures.  The  value  of  W[x]+t  was  the  sum  of  W^''\x]+t-i
and  W'-''\x]+t,  namely  the  withdrawals  in  the  tth  and  (t  +  l)th
policy  years  that  were  taken  as  withdrawing  at  integral  duration
t.  Similarly  Tix]+t  represents  the  sum  of  ^(^[xi+t-i  and  T^''\x]+f
The  symbol  Gix]+t  was  used  to  represent  the  net  movement,
i.e.,  those  entering  the  exposures  at  duration  t  less  those  emerging.
Therefore  Gix]+t  =  cui+t  —  {d[xu-t-i  +  wm+i  +  T[x]+i  +  eix]+t)
and,  as  a[x]  was  used  to  represent  n[x]  so  that  G^x]  =  o-[»i  —  it>[ij,

Eix]+t  =  E[x]+t-i  +  Gix]+t  or  Eix]+t  =  IC^w+t-

A  specimen  working  sheet  is  shown  on  page  37.

This  table  contains  the  data  only  for  "New"  assurances,
namely,  issues  from  1863  on,  and  therefore  (rix]+t  has  a  value
only  at  duration  0,  representing  new  issues.

In  combining  the  select  data  for  the  construction  of  an  aggre-
gate table  it  was  necessary  to  sum  the  several  values  for  the  same
attained  age.  Thus  Wx  =  W[o]+x  +  if [u+x-i  +  W(2i+x-2  +  •  •  •
+  W[x-i]+i  +  W[x]+o  and  similarly  for  6x,  <Tx,  Tx  and  e^.  Then
Gx  =  <Tx-  (dx-i  +  Wx-}-  Tx-h  ex).

The  corresponding  working  sheet  is  given  on  page  38.

For  the  construction  of  truncated  aggregate  tables  the  follow-
ing equation  was  used,

^^(«)  =  Ex  —  iE[x]  +  E[x-i]+i  +  •  •  •  +  Eix-t+i]+t-i),

where  Ex^^^  indicates  the  exposures  at  attained  age  x  excluding
the  first  t  years  of  duration.

The  American-Canadian  Mortality  Investigation  furnishes  an
interesting  example  of  a  working  sheet  combining  policies
issued  both  before  and  during  the  period  of  investigation.  The
experience  was  based  on  amounts  insured  to  the  nearest  hundred
dollars.     Mean  durations  were  used  (see  page  39).

The  years  of  issue  shown  in  column  (2)  apply  to  the  Existing
in  column  (3).  Similarly  column  (7)  refers  to  the  Entered  in
column  (8)  These  years  were  inserted  to  make  the  tabulation
as  simple  as  possible.  Column  (10)  was  obtained  by  summing
column  (9)  from  the  bottom  up  instead  of  working  from  the  top
down  in  the  usual  manner  by  subtraction.  The  text  of  this
investigation  gives  the  formula

^[;r]  +  n   =   E[x]+n-l  —   (e  +  W  +  d   —  s)[x]  +  n,

FROM   THE   RECORDS   OF   INSURED   LIVES.

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FROM   THE   RECORDS   OF   INSURED   LIVES.

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40  CONSTRUCTION    OF  MORTALITY   TABLES

but  the  arithmetical  work  followed  the  equation

E[x]+n-i  =  Eix]+n  +  {e  +  w  -\-  d  —  s)[x]+«.

In  these  equations  d(x]+n  refers  to  the  deaths  in  the  nth  insurance
year,  and  not  to  the  deaths  at  curtate  duration  n.

Mr.  W.  J.  H.  Whittall  (J.  I.  A.,  Volume  XXXI,  167)  intro-
duced a  convenient  notation  to  show  in  any  formula  for  the
exposed  to  risk  what  methods  were  used  in  determining  the  age
at  entry  and  the  duration.     Thus

X  =  Exact  age
{x}  =  Age  next  birthday
(x)  =  Nearest  age
I  a;  I  =  Mean  age
t  =  Exact  duration
t\  =  Curtate  duration
(t)  =  Nearest  duration
\t\  =  Mean  duration

To  the  age  symbols  might  be  added  ^j  to  indicate  the  com-
pleted age,  or  age  last  birthday,  so  that    {x]  =  x\  +1.

This  notation  is  appropriate  for  aggregate  exposures,  and  for
calendar  year  and  life  year  as  well  as  policy  year  investigations,
but  a  difliculty  arises  in  applying  the  age  symbols  to  select
notation  where  the  age  at  entry  is  indicated  by  [x].  This  was
recognized  by  the  author  who  stated

"It  will  be  understood  that  in  dealing  with,  for  instance,  age  at  exit  {a;}
+  |f|,  each  symbol  is  variable,  and  the  expression  includes  entrants  of  dif-
ferent ages  and  assured  of  different  durations,  subject  only  to  the  total  ages
attained  being  correct  according  to  the  tenns  of  the  formula."

Even  in  select  notation,  however,  the  duration  symbols  may  be
used.

B.  Calendar  Year  Method.

The  calendar  year  method,  as  previously  stated,  is  not  directly
applicable  to  the  construction  of  select  tables.  In  considering
the  formula  for  the  exposed  to  risk  in  an  aggregate  table,  it  is
essential  to  define  the  symbols  carefully.

Let  Ux  =  New  entrants  aged  x  on  January  1st  of  the  year  of
entry.
Cx  =  Existing  at  age  x  on  December  31st  of  the  last  year
covered  by  the  investigation.

FROM   THE   RECORDS    OF   INSURED   LIVES.  41

Wx  =  Withdrawals  in  calendar  year  beginning  at  age  x.
6x  =  Deaths  in  calendar  year  beginning  at  age  x.
<Tx  =  Entering  observations  at  age  x  on  January  1st  of  the
first  year  covered  by  the  investigation.

Let  it  further  be  assumed  that  an  investigation  of  the  data
justifies  the  use  of  the  fraction  /  as  the  part  of  a  calendar  year
elapsed  at  the  average  date  of  entry,  and  the  fraction  g  as  the
average  part  of  a  calendar  year  elapsed  at  the  date  of  withdrawal.

Then

X  XX  X  x  —  \

Ex  =  Hnx  -  frix  +  ^<Tx  -  ^Wx  +  gwx  -  ^Bx  -  ^dx,

0  0  0  0  0

Ex+1  =  Ex-\-  rix+i  —  /(wj+1  —  nx)  +  Cx+i  —  Wx+i

+  giwx+i  —  Wx)  —  ex+1  —

(4)

It  is  usual  to  assume  that  /  =  gf  =  |,  in  which  case  this  equa-
tion is  reduced  to

F  -    JP     M  ^»+l  +  '^'  M  rr  «^»+l  +  Wx  ,.>.

iLx+i  =  iix  T 2 ^  *^*+i 2 "^"^  ~  ^

If  a  truncated  table  were  to  be  constructed,  eliminating  the
experience  of  an  integral  number  of  years,  t,  from  issue,  the  same
equations  would  apply;  but  the  age  at  entry  would  he  x  -{-  t
instead  of  x  and  all  policies  terminated  for  any  cause  at  the  end
of  t  years  or  before  would  be  eliminated  from  the  experience.

When  the  age  at  entry  is  taken  as  the  nearest  age  or  mean
age  at  issue,  the  exposures  and  deaths  are  given  for  half  ages
instead  of  integral  ages  as  mentioned  in  Chapter  III.  Thus
Hx  could  be  described  as  the  new  entrants  aged  a;  +  ^  at  date  of
issue,  a;  +  i  being  integral.  Equations  (4)  and  (5)  are  still
applicable,  but  the  ages  for  all  the  symbols  must  be  taken  as
half  ages.

In  forming  the  H^  table  the  exposed  to  risk  were  calculated
in  the  following  manner  :

Ex  =  Ex-i  +  U^x  +  rix+i)  —  liwx  +  Wx+i)  —  (dx  +  ex).

Where*  nx  =  Entrants  at  age  x  next  birthday.

*  As  the  experience  covered  all  policies  from  their  dates  of  issue,  there
were  no  cases  entering  observation  after  issue.

42

CONSTRUCTION   OP  MORTALITY  TABLES

Wx  =  Withdrawals  at  age  x  next
birthday

dx  =  Deaths  at  age  x  next  birth-
day

e,  =  Existing  at  age  x

i.e.,  age  at
entry  next
birthday  plus
mean  dura-
tion =  age
at  end  of  cal-
endar year  of
exit

This  formula  assumes  that  the  observations  close  at  the  end  of
a  calendar  year.

In  investigating  the  data  of  the  Scottish  offices  which  con-
tributed to  the  Institute  Experience,  Mr.  Meikle  transformed
this  formula  as  follows:

E.

i,nx  -  i.{dx  +  Wj:  +  ex)  +  K^x-i-i  -  'Wxfi)

=  Zw,  -  E/x  +  M^x+i  -  W^x+l)

where  /x  =  dx  +  Wx  +  e^.  Mr.  Ryan  (J.  7.  A.,  XXVI,  257)
points  out  that  Mr.  Meikle's  form  is  easier  to  apply  than  the
Institute  form,  but  suggests  a  further  modification  to  the  follow-
ing:

Ex  =   Z(Wx   -  /x)   +  K^x+l   -   Wx+l)'
0

This  would  save  one  step  in  the  calculations.

It  should  also  be  kept  in  mind  that  the  data  entering  into  this
experience  were  separated  by  age  at  entry,  in  order  that  the
effects  of  selection  might  be  seen  and  approximate  select  tables
prepared  at  a  later  date.  The  following  illustration  will  show
the  arrangement  of  the  figures :

*  Current  Age  at  Entry  30.

Number  of  Entrants  6791.

Current  Age
at  Exit.

Existing.

Discontinued.

Died.

30
31
32
33

319
252
230
235

75
365
220
153

4
28
35
49

*Current  Age  means  age  next  birthday.

FROM   THE   RECORDS   OP   INSURED   LIVES.  43

The  calendar  year  of  entry,  called  "Year  0,"  gives  the  exposed
to  risk  for  age  29.  The  next  calendar  year,  or  "Year  1"  refers
to  age  30  and  so  on.  Thus  by  the  Institute  formula  for  this  one
age  at  entry

£729  =^-y=  2858,

E^o  =  ^29+  ^  -  Z5_+365  _^^_^  3jg^  ^  ^210.5.

In  forming  the  H™  table  the  figures  were  combined  for  each
attained  age,  irrespective  of  age  at  entry.

C.  Life  Year  Method.

As  the  life  year  method  is  not  suitable  for  the  construction  of
select  tables,  consideration  will  be  given  only  to  aggregate
tables.  Let  us  suppose  that  exact  ages  at  entrance  and  with-
drawal have  been  tabulated.

Then  let  rix  =  New  entrants  at  age  x  last  birthday.
Wx  =  Withdrawals  at  age  x  last  birthday.
6x  =  Deaths  at  age  x  last  birthday.

Let  it  also  be  assumed  that  the  aggregate  of  the  fractions  of
a  year  since  last  birthday  at  the  time  of  entry  of  n^  are  tabulated
and  denoted  by  /«,  and  that  Qx  is  the  sum  of  the  fractions  of
exposures  in  the  year  of  withdrawal  among  Wx.

Then

X  X  x—l

Ex  =  T,nx  -  fx  -  T.Wx  -\-  9x  -  Z^x  ra\

0  0  0  \p)

Ex+1  =  Ex  +  rix+i  —  (fx+i  -  fx)  -  Wx+i  +  (gx+i  -  gx)  -  Qx-

If  instead  of  using  fx  and  gr,,  which  are  the  sums  of  the  fractions
and  vary  with  the  attained  age,  average  fractions  per  entrant  and
withdrawal,  /  and  g,  are  adopted  for  all  ages,  we  have

X  X  x—l

Ex  =  T.nx  -  fux  -  T.Wx  +  gwx  -  Z^x,

0  0  0

Ex+1  =  Ex  +  rix+i  —  fiux+i  —  rix)  —  Wx+i

+  giwx+i  -  Wx)  -  dx
=  Ex+  {fux  +  (1  -  /)n^i}

-  {gwx-i-  {I  -  g)wx+i}  -  Ox.

If  there  are  persons  under  observation  when  the  period  of
exposure  began  and  existing  at  the  end  of  that  period,  the  former

(7)

44  CONSTRUCTION   OF  MORTALITY   TABLES

may  be  treated  as  new  entrants  at  their  ages  at  which  they  came
into  the  experience  and  the  latter  in  the  same  manner  as  with-
drawals. In  such  a  case  they  must  of  course  be  taken  into
account  in  arriving  at  the  average  fractions  /  and  g.

It  has  been  stated  in  Chapter  III  that  the  ages  at  entry  and
withdrawal  may  be  taken  as  the  mean  ages.  It  might  be  con-
tended that  we  then  have  a  calendar  year  method.  This  is  not
the  case,  however,  as  under  the  latter  method  the  deaths  are
analyzed  by  calendar  years,  i.e.,  by  age  at  the  beginning  of  the
calendar  year  of  death.  Under  the  life  year  method  the  deaths
are  analyzed  by  the  age  last  birthday  at  death,  and  mean  ages
at  entry  and  withdrawal  are  used  simply  as  a  device  for  getting
rid  of  fractional  exposures  at  such  times.*

If  the  observations  begin  and  close  with  persons  under  ob-
servation or  existing,  their  ages  at  such  times  could  not  or-
dinarily be  taken  as  mean  ages.  The  age  nearest  birthday
might  be  used.  When,  however,  such  exposures  commence
and  cease  at  the  end  of  two  different  calendar  years,  another
convenient  method  may  be  adopted.  Let  <Tx  represent  those
entering  on  January  1st  at  the  start,  at  mean  age  x  in  the  pre-
ceding calendar  year.  They  will  be  exposed  for  half  a  year
at  age  x  as  we  may  assume  that  on  the  average  they  are  aged
X  +  I  on  January  1st,  when  the  exposures  begin.  Similarly
Bx  represents  the  existing  on  the  final  December  31st,  age  x
being  the  mean  age  in  the  year  just  ended.  There  will  there-
fore be  on  the  average  a  half  year  of  exposure  at  age  x.  Then
if  X  represents  the  mean  age  in  the  symbols  Ux  and  Wx,  while  6x
represents  the  deaths  at  age  x  last  birthday,  we  have

X  X  X  —  1  X  X

Ex   =    Z^x  +    Z<^s   —    h<^x   -    Z^x   -    T.^x   —    T.^x  +    h^x,

0  0  u  0  0

E^+1  =  Ex^-  Wx+1  +  hicx  +  cx+i)  -  dx  (8)

-  Wx+1  -  Kcx  +  ex+i).

If  the  period  of  investigation  had  not  extended  over  exact
calendar  years,  equation  (8)  could  still  be  used  if  x  in  the  symbols
(Tx  and  Cx  were  the  age  last  birthday,  for  we  should  still  have
approximately  one  half  year  of  exposure  at  age  x  in  both  cases.

*  Mr.  W.  J.  H.  Whittall  (J.  7.  A.,  Vol.  XXXI,  163)  discusses  the  definition  of
methods  and  states  (p.  164):  "It  would,  therefore,  seem  probable  that  in  the
system  of  determining  the  ages  at  death  in  any  particular  formula  will  be
foimd  the  key  to  its  proper  classification."

FROM   THE   RECORDS   OF   INSURED   LIVES.  45

The  last  method  of  estimating  exposures  mentioned  in  Chapter
III  is  to  take  the  ages  at  entry  and  exit  as  the  ages  nearest
birthday,  except  for  the  age  at  death.  This  produces  no  frac-
tional exposures,  as  those  under  observation  at  the  start  or
existing  at  the  close  of  the  period  may  be  likewise  treated  and
included  in  the  new  entrants  and  withdrawals  respectively.

Then

X  r— 1  X

0  0  0  \^)

Ex+1  =  -EJx  +  n^+i  -  dx  -  W:,+i.

One  example  of  the  life  year  method  is  the  experience  of  the
Economic  Office,  1862,  taken  out  by  J.  J.  Downes,  who  was  the
introducer  of  the  card  system.  A  description  of  the  methods
employed  is  given  by  W.  J.  H.  Whittall,  J.  L  A.,  Vol.  XXXI,  171.
Exact  ages  to  four  decimal  places  were  computed  at  entrance  and
exit,  which  is  an  unnecessary  refinement.  The  method  of
tabulation  was  also  laborious.

Mr.  A.  J.  Finlaison  used  the  life  year  method  in  compiling  the
Government  Annuitants'  Experience,  1883.  After  a  test  he
assumed  that  on  the  average,  four  months  had  elapsed  at  date
of  entry  since  the  last  birthday.  Accordingly  each  new  entrant
was  assumed  to  be  exposed  to  risk  for  two-thirds  of  a  year  at
the  age  last  birthday  at  entry.  Ages  at  exit  were  taken  ac-
curately, the  exposures  on  the  existing  terminating  on  exact
birthdays.  Mr.  Finlaison  used  the  data  to  form  select  tables  of
annuitants'  mortality  by  assuming  that  the  eight  months  mor-
tality rate  following  entry  was  the  rate  for  the  first  year  after
entry,  and  so  on.  Thus  the  approximate  periods  of  experience
0-f,  f-lf,  11-21,  etc.,  were  considered  to  be  annuity
years  1,  2,  3,  etc.  This  illustrates  how,  in  attempting  to  get
mortality  rates  for  each  exact  year  of  life,  the  advantages  of
accurate  tables  by  policy  years  must  be  sacrificed.  As  the
effect  of  self-selection  by  annuitants  is  often  marked  for  the
first  few  policy  years,  the  understatement  of  the  mortality  in
the  early  policy  years  under  Mr.  Finlaison's  method  would
appear  to  be  more  important  than  the  loss  of  accuracy  that  would,
be  introduced  by  the  use  of  the  policy  year  method.  The
annuity  experience  of  the  British  Offices,  1863-1893,  was  based
on  annuity  years,  the  age  at  entry  being  the  age  nearest  birthday
at  date  of  purchase.

Chapter  VI.    Tables  with  More  than  One  Decremental

Factor.

A  mortality  table,  as  heretofore  considered,  is  a  table  showing
the  number  living  at  each  age,  either  in  the  select  or  in  the  aggre-
gate form.  This  number  decreases  as  the  age  increases  only
because  of  the  deaths.  Death  is  the  only  decremental  factor
in  such  a  table.  We  may,  however,  have  a  table  which  has  one
or  more  additional  decremental  factors,  such  as  those  of  with-
drawal, marriage  or  the  re-marriage  of  widows.  If  the  addi-
tional factor  is  that  of  withdrawal,  then  the  table  will  show  for
each  age  the  number  living  who  have  not  withdrawn,  and  in
proceeding  from  one  age  to  the  next  there  is  deducted  the  number
who  have  withdrawn  and  those  who  have  died  without  with-
drawing during  that  year  of  age.  There  is  given  below  a  speci-
men table  (J.  /.  A.,  Vol.  XXXIII,  196)  in  the  select  form  for
age  20  at  entry:

Mortality  Table,

Showing  Mortality  and  Withdrawal.

Deaths

Dura-

Numbers

Deaths.

With-

and

tion.

Living.

drawals.

With-
drawals.

(0.

^M+t-

<^W+<-

WM+t.

(d  +u>)(xi+i.

0

100,000

426

14.970

15,396

1

84.604

373

10,549

10,922

2

73,682

335

7,717

8,052

3

65,630

307

5,892

6.199

4

59,431

297

4,741

5,038

Before  starting  to  analyze  the  experience  from  which  it  is  planned
to  construct  such  a  table  we  must  examine  the  functions  by
which  we  may  compute  the  successive  figures  in  the  table.  The
rate  of  withdrawal  will  ordinarily  vary  with  the  duration  of
insurance  as  well  as  with  age  and  therefore  select  tables  are
best  adapted  to  display  the  results.  It  may  also  be  noted  that
the  rate  usually  varies  with  the  plan  of  insurance,  making  sep-
arate tables  by  plan  desirable  in  many  instances.  The  investi-
gation might  be  based  on  lives,  policies,  or  amounts  insured.

46

FROM   THE   RECORDS    OF   INSURED   LIVES.  47

Let  US  adopt, the  following  notation:
Table  Notation:

l'ix]+t  =  the  survivors  at  age  x  -}-  t  of  the  entrants  at  age

X  who  have  not  withdrawn,
d'ix]+t  =  the  deaths  during  the   (t  +  l)th  year,   before

withdrawal.
w'[x]+t  —  the  withdrawals  during  the  (t  +  l)th  year.

Investigation  Notation:

Eix]+t  =  the  exposed  to  risk  of  death.
(wE)ix]+t  =  the  exposed  to  risk  of  withdrawal,  the  deaths
being    given    their    respective    fractional
durations  while  the  withdrawals  are  given
a  full  year's  exposure.
Oix]+t  =  the  deaths  in  the  {t  +  l)th  year.
W[a!]+t  =  the  withdrawals  in  the  {t  +  l)th  year.

It  may  be  noticed  that  the  definition  of  (wE)[x]+t  is  consistent
with  the  method  by  which  Eix]+i  is  determined,  as  in  the  latter
case  the  withdrawals  are  given  their  proper  fractional  exposures
but  the  deaths  are  counted  as  exposed  to  risk  for  the  entire  year
of  death.  Thus  to  find  the  proportion  that  withdraw  of  those
persons  who  are  exposed  to  withdrawal  for  a  year,  we  must
count  each  person  who  withdraws  as  a  unit  and  not  a  fraction
in  the  exposures.

In  tabulating  W[x]+t  the  question  arises  whether  the  with-
drawals at  the  exact  end  of  a  policy  year,  which  are  generally
a  large  proportion  of  the  total  withdrawals,  should  be  grouped
with  those  during  the  year  just  closed  or  with  those  of  the  sub-
sequent year.  If  we  consider  the  first  policy  year  it  will  be  seen
that  the  former  course  must  be  adopted  in  order  to  get  a  full
year  of  withdrawals  recorded  in  that  year.  We  shall  then  have
the  rates  of  withdrawal  during  and  at  the  end  of  each  year,  and
l'[x]+t  will  represent  the  survivors  of  the  original  entrants  who
continue  their  policies  beyond  t  years  from  issue.  It  should  be
remembered,  however,  that  not  all  the  l'ix]+t  persons  continue
their  policies  to  the  end  of  the  {t  +  l)th  year,  for  some  will
withdraw  at  fractional  durations.

If  we  now  consider  the  double  decrement  table  we  see  that  the

48  CONSTRUCTION   OF  MORTALITY  TABLES

probability  that  a  life  aged  [x]  +  t  will  withdraw  within  a  year  is

which  is  different  from  the  rate  of  withdrawal,  among  those
who  are  alive.     This  latter  rate  is

i\x]+t—hd'ix]+t

on  the  assumption  of  a  uniform  distribution*  of  deaths,  and  is

the  rate  given  in  an  investigation  by  the  ratio  W[x]+il(wE)[x]+f

Similarly  the   probability   of  a  life   dying   before   withdrawal

becomes

^f        _  d'ix]+t
Q  w+t  -  77 >

whereas  the  rate  of  mortality,  on  the  assumption  of  a  uniform
distribution  of  withdrawals,  is

Qlx]+t  —  77 i:;^        •

The  value  of  q[x]+t  is  given  in  the  investigation  by  dix]+t/E[x]+t'
It  is  evident  that  in  a  single  decrement  mortality  table  q'  becomes
g,  because  the  column  of  survivors,  I,  is  reduced  only  by  deaths.

In  order  to  construct  a  table  from  the  data  we  must  find  rela-
tions by  which,  starting  with  a  given  radix,  we  may  find  the
values  of  d'lxut  and  w'[x]+t-  We  can  then  construct  the  V  column
by  the  equation  I'm+i+i  =  l\x]+t  —  d\x]+t  —  «>'[*]+<.

The  simplest  procedure  would  appear  to  be  to  calculate  the
values  of  q'  and  (wq)  directly  from  the  data,  for  these  prob-
abilities when  multiplied  by  I'  give  us  d'  and  w\  If  this  were
done  we  should  find  the  probabilities  from  the  data  as  follows.

q  w+*  =  p7 —  >  UU;

{wq),xu^=j^^^^,  (11)

where  E'  and  {wE')  are  obtained  by  giving  6[x\+t  and  tftii+t  a

*  It  should  be  noticed  that  this  assumption  as  applied  to  a  double  decrement
table  is  not  equivalent  to  the  assumption  of  a  uniform  distribution  of  deaths
in  a  single  decrement  table.

FROM  THE  RECORDS  OF  INSURED  LIVES.  49

full  year's  exposure  in  both  cases,  and  therefore

E\x]+t   =    {wE')[x]+f

Using  the  notation  for  single  decrement  tables  except  for  the
change  in  the  definition  of  W[x]+t,  and  assuming  that  the  period  of
observation  begins  and  ends  on  policy  anniversaries,  we  have

t  <— 1  t  t—\  t

0  0  0  0  1

where  T[x]+t  represents  the  matured,  such  as  endowments  and
terms,  taken  to  mean  or  nearest  durations.     Then

E\x]+t+i  =  E'[x]+t  +  o'[x]+<+i

—  i'^ix]+t  +  G[x]+t  +  T[x]+t+i  +  e[x]+t+i)'        (12)

This  gives  us  the  denominator  common  to  equations  (10)  and  (11).

It  frequently  happens  that  the  probabilities  q'  and  (wq)  are  not
as  useful  as  the  rates  q  and  (wr).  If  we  wish  to  compare  the
results  of  the  investigation  with  those  of  other  experiences  the
latter  rates  should  be  used  as  q  is  independent  of  (wr)  and  vice
versa,  while  q'  and  (wq)  are  each  functions  of  both  the  mortality
and  the  withdrawal  rates.  It  may  be  desired  to  combine  the
withdrawal  rate  from  one  experience  with  the  mortality  rate  of
another,  in  which  event  we  need  to  have  the  independent  rates,
q  and  (wr).

Let  us  then  see  how  we  can  construct  the  double  decrement
table  from  these  rates.

On  the  assumption  of  a  uniform  distribution  of  deaths  and
withdrawals  during  policy  years  we  have

(V  -  W)q  =  d\
(V  -  hd')(im')  =  w'.

Solving  the  equations  we  have

^,  ^l'q[l -h(wr)\ ^  (13)

^  V(wr)[l  -  M  .  (14)

1  -  \q(wr)  ^  ^

We  may  thus  compute  the  decrements  from  the  rates  q  and
(wr),  which  can  be  obtained  from  the  deaths,  withdrawals,  and
exposed  to  risk  as  previously  shown.  We  can  also  construct
a  table  where  q  is  derived  from  one  experience  and  (wr)  from

60  CONSTRUCTION   OP  MORTALITY   TABLES

another.     We  may  also  obtain  a  check  formula  for  the  V  column,
as  follows:

l+i'  =  1'  -d'  -w'

^  „  r     _  q-h  (wr)  -  q{wr)  1

*L  \-\q{vyr)       J  (15)

^  ^,(1  -g)[l  -  {wr)]-\q{wr)
1  -  \q{wr)

These  equations  were  developed  by  Dr.  T.  B.  Sprague  in  a
paper  on  marriage  and  mortality  tables  (J.  /.  A.,  Vol.  XXI,  406)
and  were  applied  to  tables  of  mortality  and  withdrawal  by  Mr.
T.  G.  Ackland  (J.  /.  A.,  Vol.  XXXIII,  194).  Mr.  Ackland,  in
his  paper,  discusses  very  fully  the  use  of  exact,  mean  and  nearest
durations  in  obtaining  the  exposed  to  risk,  E[x\^t  and  {wE)ix\Jrti
and  concludes  that  the  nearest  duration  method  is,  upon  the
whole,  the  best  suited  for  obtaining  the  rates  of  mortality  and
withdrawal  from  a  large  body  of  insured  lives.

Mr.  Ackland  deduces  the  formulae  for  the  exposed  to  risk
where  the  period  of  investigation  extends  over  integral  policy
years  in  the  following  manner.

Let        S[ai+t  =  the  survivors  in  force  at  the  commencement
of  the   period   of  observation,   at  integral
duration  t.
n[x\  =  the  new  entrants  at  age  x.
^[x]+t  =  the  cases  existing  at  the  close  of  the  period  of
observation,     at    integral    duration    t.
{aw)[x\+t  =  the   withdrawals   having   a   duration   greater
than  t  that  are  treated  as  having  integral
duration  t,  i.e.,  those  with  a  duration  less
than  <  +  .5  and  one-half  those  having  a
duration  of  exactly  t  +  .5.
(bw)[x]+t  =  the  balance  of  the  withdrawals  with  duration
greater  than  t  but  not  in  excess  of  <  +  1.
These  are  treated  as  having  integral  dura-
tion t  -\-  1.
Then  {aw)ix]+t  +  (hw)ix]+t-i  =  Ww+t,  the  total
withdrawals  at  integral  duration  t  by  the
nearest  duration  method.
{ad)ix]+t  =  deaths  in  first  half  of  {t  +  l)th  year,

FROM  THE  RECORDS  OP  INSURED   LIVES.  61

(hd)ix]+t  =  deaths  in  last   half  of   (t  +  l)th   year,   and
letting  d[x]+t  =  {ad)[x]+i  +  (6d)[*]+<-i*

also,  let

where
Then

(e  +  M)  4-  d)M+t  =  Fir]+t
(s  —  F)[x]+t  =  G[x]+t

t  =  0,        G[x]  =  nix]  —  Fix].

Eix]+t  =  iiG)  +  (ad)ix]+t  (16)

u

(wE)ix]+t  =  JliG)  +  {aw)ix]+t  (17)

0

On  page  52  is  given  a  copy  of  the  working  sheet  used  to  carry
out  the  tabulation  for  age  20  at  entry.

Mr.  Ackland's  notation  is  retained  in  this  table.  It  must  be
carefully  noted  that  W[x]+t  in  columns  (4)  and  (13)  are  not  the
same  functions,  and  a  like  condition  is  true  of  dix]+t  shown  in
columns  (5)  and  (10).  Column  (11)  is  obtained  by  dividing
(10)  by  (9),  and  column  (14)  is  the  quotient  of  (13)  -4-  (12).
The  brackets  in  column  (5)  indicate  how  the  numbers  in  columns
(4)  and  (5)  were  combined  to  obtain  (6).  Columns  (10)  and
(13)  were  obtained  from  (5)  and  (4)  respectively  by  combining
the  figures  set  down  opposite  each  duration.

The  data  being  analyzed  had  been  obtained  from  a  Clerks'
Association  in  which  subscriptions  were  payable  monthly  on
the  first  day  of  each  month.  Therefore  the  withdrawals  could
be  assumed  to  be  distributed  uniformly  over  each  year  of  dura-
tion. Mr.  Ackland  found  that  the  exact  average  fractional
duration  at  withdrawal  was  approximately  one  half  year.  Ac-
cordingly formulae  (13),  (14)  and  (15)  can  be  considered  suffi-
ciently accurate.

Dr.  Sprague,  in  the  paper  previously  mentioned,  also  developed
for  a  double  decrement  table  central  rates  corresponding  to

m

^        2       2

*  It  will  be  seen  that  rf[i]+«  has  not  the  usual  significance  of  6ix]+t  or  d[x]+t
but  is  calculated  similarly  to  wix]+t  in  accordance  with  nearest  durations.

62

CONSTRUCTION    OF   MORTALITY  TABLES

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FROM   THE   RECORDS   OF   INSURED   LIVES.  53

and

(wm)  =

w
*        2       2

which  are  mutually  independent  but  from  which  can  be  computed,
if  desired,  the  rates  q  and  (wr)  by  the  relations

2m

3  =

2  +  m

and

/,„.^  _     2(w;m)
(_wr)  =

2  +  (wm)

Central  rates  were  employed  by  Mr.  Ackland,  who  shows  that

m

and

and  also  that

d'  =  V

w'  =V

m  +  (wm)
"^  2

(«jm)

^       w  +  (i«wi)

1  -

2

w  +  (rum)

1       w  +  (ww)

In  computing  the  central  rates  he  divided  6[x]+t  and  W[x]+«  by
the  adjusted  exposed  to  risk,  E[x]+t,  in  each  case.  Eix]+t  was  the
number  exposed  to  risk  computed  up  to  the  actual  cessation  of
the  risk  from  any  cause — under  the  nearest  duration  method,
to  the  nearest  integral  year.  This  makes  it  unnecessary  to
calculate  separate  exposed  to  risk  for  deaths  and  withdrawals.
These  equations  are,  however,  incorrect  where  there  is  not  a
uniform  distribution  of  withdrawals.

A  life  insurance  company  which  issues  policies  only  with
annual,  semi-annual  and  quarterly  premiums  could  not  assume
a  uniform  distribution  of  withdrawals.  For  example,  assuming
that  lapse  is  as  likely  at  the  second  semi-annual  premium  as  at
the  first,  the  average  fractional  duration  would  be

Ks*'

)-l

64  CONSTRUCTION   OF  MORTALITY  TABLES

and  similarly  for  quarterly  lapses

4\4^2^4^     /      8

If  we  assume  62^  per  cent,  of  the  lapses  are  on  policies  issued  at
annual  rates,  32^  per  cent,  at  semi-annual,  and  5  per  cent,  at
quarterly,  the  average  fractional  duration  for  all  lapses  would  be

.625  +  .325  X  7  +  .05  X  i  =  .9.
4  8

If  we  felt  warranted  in  considering  that  this  represented  the
proper  average  fraction  for  all  withdrawals,  including  surrenders,
at  all  durations,  we  might  record  W[x]+t  in  column  (4)  as  equal
to  .9wix]+t-i  +  .lw[x]+t,  i.e.,  refer  nine-tenths  of  the  withdrawals
during  or  at  the  end  of  any  year  to  the  integral  duration  at  the
end  of  the  year,  and  one-tenth  to  the  beginning  of  the  year,
instead  of  tabulating  by  the  nearest  duration  method  the  func-
tions (aw)  and  (bw).

This  example  points  out  the  error  in  assuming  in  such  a  case
that

{l'-l«')i-d'.

Continuing  the   above  illustration   it   would   be   necessary   to
change  this  equation  to

{V  -  .lw')q  =  d',

which,  together  with

(r-id')(«;r)  =w',

might  be  solved  to  give  d'  and  w'  in  terms  oil',  q  and  {wr).

In  the  discussion  of  Mr.  Ackland's  paper,  Mr.  R.  Todhunter
presented  a  new  view  of  v/ithdrawal  rates.     He  states :

"I  suggest,  in  the  first  place,  that  discontinuances  in  ordinary  business  are
attributable  to  a  force  essentially  different  in  character  from  the  force  which
operates  to  produce  mortality.  Death  claims  are  caused  by  a  force  which
operates  continuously  throughout  the  history  of  policies,  whereas  discon-
tinuances are  caused — so  far  as  regards  the  large  majority  of  policies — ^by
a  discontinuous  force  coming  into  operation  at  certain  recurring  epochs.  A
pohcyholder  does  not  exercise  his  option  of  withdrawing  continuously  in  the
same  sense  that  he  is  continuously  subject  to  the  risk  of  death;  he  exercises  it
periodically,  in  most  cases  on  the  occasions  of  his  receiving  a  renewal  notice,

FROM   THE    RECORDS    OF    INSURED    LIVES.

55

and  having  to  decide  whether  to  pay  or  not  to  pay.  This  applies,  I  think,
to  nearly  all  lapses,  the  only  exceptions  that  occur  to  me  being  the  few  cases
in  which  a  poUcyholder  pays  a  series  of  fines  to  extend  the  days  of  grace  for
successive  short  periods;  and  lapses  constitute  a  very  large  proportion  of
the  whole  number  of  discontinuances.  It  applies,  also,  to  all  surrenders
carried  out  at  or  about  the  renewal  date.  The  proportion  which  these  form
of  the  entire  body  of  surrenders,  will  no  doubt  vary  in  different  classes  of
business;  in  a  small  experience  of  two  years,  I  found  that  55  out  of  a  total  of
94  surrenders  took  place  during  the  days  of  grace,  11  took  place  within  a
month  before  the  renewal  date  (probably  on  receipt  of  renewal  notices),  and
the  remaining  28  were  scattered.  Having  regard  to  the  small  proportion
that  the  number  of  scattered  surrenders  forms  of  the  entire  number  of  lapses
and  surrenders,  I  think  it  may  fairly  be  stated  as  a  general  proposition,  that
discontinuances  are  mainly  due  to  the  exercise  of  a  periodical  option  at  or

"If  this  proposition  be  admitted,  it  follows  that  the  force  which  causes
discontinuances  would  be  more  appropriately  measured  by  rates  of  non-
renewal, than  by  rates  or  forces  of  withdrawal — in  other  words,  by  the  ratios
that  the  withdrawals  at  definite  epochs  bear  to  the  exposed  to  risk  of  with-
drawal at  those  epochs,  than  by  the  ratios  that  the  withdrawals  in  given
periods  bear  to  certain  numbers  supposed  to  be  continuously  exposed  to  the
risk  of  withdrawal  throughout  those  periods.  As  applied  to  a  collected  ex-
perience, this  second  proposition  presupposes  a  pohcy-year  tabulation  of  the
observed  facts.  A  tabulation  by  calendar-years,  or  years  of  life,  will,  of
course,  have  the  effect  of  spreading  the  discontinuances  over  the  years  of
observation,  and  will  thus  exhibit  something  of  the  nature  of  a  continuous
force  of  discontinuance.  If  it  be  admitted  that  such  a  force  has  no  real  ex-
istence, the  fact  that  it  is  artificially  created  by  any  method  of  tabulation
other  than  one  that  follows  the  years  of  assurance  may  be  considered  another
argument  in  favour  of  the  PoUcy-year  Method."

The  sort  of  table  suggested  can  be  clearly  seen  from  that
given  below.  *

Year

Exposed  to

Exposed  to

Rate  of

of

Risk

d.

Q-

Risk  of  Non-

w.

Non-

Aasurance.

of  Death.

Renewal.

Renewal.

1

8,016

30

.0037

7,986

1,288

.161

2

6,313

36

.0057

6,277

476

.076

3

5,535

25

.0045

5,510

297

.054

4

4,945

29

.0059

4,916

149

.030

5

4,404

25

.0057

4,379

176

.040

6

3,958

15

.0038

3.943

95

.025

Total

33,171

160

.0048

33,011

2,481

.075

The  nearest  duration  method  is  a  convenient  one  for  arranging
the  withdrawal  data  in  the  proper  form  for  determining  the  rate

*  Terminations  other  than  by  death  and  withdrawal  are  not  shown  in  this
table.

56  CONSTRUCTION    OF   MORTALITY   TABLES

of  non-renewal.  Each  of  the  numerous  withdrawals  at  the
end  of  the  policy  year  would  be  located  exactly.  Those  at
semi-anniversaries  would  be  alternately  located  at  the  beginning
and  end  of  the  year,  lapses  at  the  end  of  the  first  and  third
quarters  would  be  thrown  to  the  nearest  anniversary  as  would
also  the  usually  small  proportion  of  surrenders  that  are  scattered
through  the  year.  The  result  would  be  that  all  withdrawals
would  be  considered  as  taking  place  on  policy  anniversaries.
This  distorts  the  facts  somewhat,  but  on  the  whole  gives  a
more  nearly  correct  view  than  to  assume  that  all  the  withdrawals
are  spread  throughout  the  policy  year.  Mr.  Todhunter  sug-
gests that  it  may  be  desirable  in  a  large  experience  to  exhibit
the  rates  of  mortality  and  non-renewal  by  quarters  of  a  year
for  the  first  two  policy  years.

It  may  also  be  seen  that  a  double  decrement  table  constructed
in  this  manner  is  a  useful  one  for  determining  the  present  value
of  future  commissions,  because  V[x]+t  will  be  the  number  of
survivors  at  duration  t  who  pay  premiums  to  the  end  of  the
(t  4-  l)th  year.  The  table  could  also  be  used  for  comparative
rates  of  non-renewal  (J.  I.  A.,  Vol.  XXXIII,  277,  278).

In  obtaining  the  exposed  to  risk  and  rates  of  mortality  and
non-renewal  we  may  in  general  record  the  data  as  shown  in
Chapter  V  by  the  policy  year  and  nearest  duration  methods  for
a  single  decrement  table,  but  we  must  separate  the  matured
from  the  withdrawals.  Furthermore  we  must  include  in  <Tix]+t
all  policies  terminated  prior  to  the  period  of  investigation  at
nearest  duration  t  and  those  terminated  at  exact  duration  t.
These  additional  policies  will  be  included  also  in  W[x]+t  so  that
they  will  both  enter  and  also  withdraw  at  duration  t.  This
will  have  no  effect  on  the  exposed  to  risk  of  death  but  will  give
the  necessary  data  for  the  rate  of  non-renewal  at  duration  t.
If  <r[x]+t  were  not  to  include  these  cases,  such  entrants  would
have  to  be  excluded  from  the  exposed  to  risk  of  non-renewal
at  duration  t,  making  it  necessary  also  to  exclude  the  withdrawals
at  durations  between  t  and  t  -\-  .5  among  such  entrants.  This
could  be  done  by  excluding  from  o-[x]+t  all  cases  terminated  by
withdrawal  at  durations  less  than  t  -\-  .5  and  half  of  the  with-
drawals at  that  exact  point — as  this  would  not  affect  Eix]+t — and
by  excluding  <Tix]+t  from  the  exposed  to  risk  of  non-renewal  at
duration  t.  Let  us  assume  that  the  former  course  has  been

FROM   THE   RECORDS    OF   INSURED    LIVES.  57

Similarly  for  the  existing,  let  us  suppose  that  eix]+t  excludes
policies  withdrawn  at  nearest  duration  t  even  though  they  may
have  withdrawn  after  the  policy  anniversary,  and  that  the
policies  so  excluded  are  included  in  W[x]+«-  Then  eix]+t  will  be
included  in  the  exposed  to  risk  of  non-renewal  at  duration  t.

If  we  represent  the  matured  at  nearest  duration  t  by  T[xutj
formula  (3)  for  the  exposed  to  risk  at  death  will  become

E[x]+i+i  =  E[x]+i  +  <rix]+t+i

—    C^lxl+t+l  +  Wix]+t+l  +  ^M+t+l  +  d[x]+t)  (18)

and

Qlx]+t  =

E

[*]+<

the  rate  of  mortality  and  also  the  probability  of  death  in  the
double  decrement  table.    Also  (nrE)[x]+t,  the  exposed  to  risk
of  non-renewal  at  the  end  of  the  (t)th  year,
t  t

t-i  t-i  <— 1

—  zl  "^M+t  —  zL  ^w+«  —  zl  ^i']+t-

Then

(nrE)  [x]+t+i  =  inrE)[x]+t  +  <Tix]+t+i

—    (T[x]+t+l  +  W[x]+t  +  €[x]+t  +  d[x]+t)
=   Eix]  +  t  +  <Tlx]+t+l   —    T[x]  +  t+l   —  dix]  +  t'

If  then  we  let

fix]+t  =  wix]+t  +  eix]+t
and

9M+t  —  <^{x\+t  —  T[x\j^t  —  ^[i]+<-i
we  have

E[x\Jrt+i  =  Eix]+t  +  gix]+i+i  —  fix]+t+i  (19)

and

(nrE)[x]+t+i  =  E[x]+t  +  gM+t+i-  (20)

The  rate  of  non-renewal  at  the  end  of  the  (<)th  year  is

Wlx]+t

(nrq)

lx]+t   —

(nrE)

lx]+t

Having  calculated  the  rates  qix]+t  and  {nrq)ix]+t  we  can  con-
struct the  double  decrement  table  by  the  relation

l'lx]+t+i  =  (l'ix]+t  —  q[x]+t  l'[x]+i)[l  —  (nrq)ix]+t+i]

=  l\x]+t(l  -  qix]+t)[l  -  (nrq)[x]+t+i].  (21)

58  CONSTRUCTION   OF   MORTALITY   TABLES

This  form  of  the  equation  shows  that  the  probabilities  of  living
and  of  renewal  are  entirely  independent  of  each  other.

The  data  which  are  used  as  the  basis  of  the  rates  of  mortality
and  withdrawal  may  include  policies  issued  with  deferred  dividend
periods.  Such  policies  were  so  often  sold  with  emphasis  on  the
total  cash  surrender  value  available  at  the  end  of  the  dividend
period,  that  it  may  be  considered  advisable  to  treat  them  as
matured  on  the  dividend  due-date.

Another  point  which  may  be  mentioned  is  that  it  may  be
found,  in  any  particular  experience,  that  the  non-renewal  or
withdrawal  rate  may  vary  with  duration  only  for  a  certain
number  of  years.  The  rate  might  even  become  practically
constant  for  all  attained  ages  beyond  a  fixed  duration.  In
either  case  tables  of  mortality  and  withdrawal  could  be  cal-
culated in  the  select  and  ultimate  form,  the  duration  of  the
select  period  being  the  greater  of  the  select  period  of  mortality
and  the  select  period  of  withdrawal.

Tables  of  mortality  and  withdrawal  have  been  the  commonest
forms  of  double  decrement  tables  based  on  the  records  of  in-
sured lives  in  America,  but  in  recent  years  tables  having  de-
crements from  mortality  and  total  and  permanent  disability
have  assumed  great  importance.  The  probabilities  involved
together  with  formulae  for  the  exposed  to  risk  are  given  in
Actuarial  Studies  No.  5.

In  Great  Britain  the  subject  of  marriage  and  mortality  has
been  of  importance  to  actuaries  because  of  such  contingencies
that  have  been  occasionally  insured  against  by  the  British
Companies.  Dr.  Sprague's  paper  presented  in  1879  {J.  I.  A.,
Vol.  XXI,  406)  has  already  been  mentioned.  His  formulae,
developed  on  the  assumption  of  a  uniform  distribution  of  deaths
and  marriages  throughout  the  year,  are  applicable  to  that  sub-
ject, though  introducing  a  considerable  error  when  used  for
withdrawals  in  ordinary  insurance  experience.  In  a  later  paper
(J.  /.  A.,  Vol.  XXVIII,  350)  Dr.  Sprague  has  given  tables  for
the  re-marriage  of  widowers,  showing  the  necessity  for  a  select
period  for  "recent  widowers"  merging  into  an  ultimate,  or
"chronic  widowers"  rate  at  the  end  of  twenty  years  for  those
who  become  widowers  at  age  25,  the  select  period  shortening
as  the  age  increases.  In  a  paper  in  the  same  volume  (p.  384)
Mr.  J.  Chatham  brings  out  that  in  constructing  the  select
portion  of  such  tables  we  may  obtain  ^{wl)^x\+t,  the  widowers,

J

FROM   THE    RECORDS   OF   INSURED  LIVES.  69

becoming  such  at  age  x,  who  are  alive  and  have  not  re-married
at  age  x  ■\-  t,  by  working  backward  from  (m>Z)[xi+«+i  by  the
relation

where  m  and  m  are  respectively  the  central  marriage  and  death
rates  at  age  [x]  +  t,  and  similarly  for  the  individual  decrements :

(«?m)[xi+t,  widowers  marrying,   =  {wl)[x\+t+\-  ^  _  ir — X — ]
{wd)[^]+t,  widowers  dying,   =  (w;Z)[xi+«+i-  ^  _  rr^^  ^     \ '

These  equations  are  obtained  by  transforming  the  equations
corresponding  to  those  contained  on  p.  53.

Messrs.  Hewat  and  Chatham  investigated  the  mortality  and
marriage  experience  of  the  Widows'  Funds  of  the  Scottish  Banks
(J.  /.  A.,  Vol.  XXXI,  428)  by  calculating  the  central  rates.
The  period  of  investigation  was  closed  for  the  different  banks
in  such  a  way  as  to  lead  to  the  assumption  that  the  existing  were
exposed  to  risk  on  the  average  for  six  months  after  the  age  at
which  they  were  recorded  as  terminating.  They  describe  their
method  as  being  analogous  to  the  exact  duration  method.
Using  (6m) J  to  indicate  the  number  of  bachelors  marrying  be-
tween the  ages  x  and  x  -\-  1  and  {hd)x  to  indicate  the  bachelors
dying  within  that  year  of  age,  they  employed  the  following
formula  for  the  number  of  bachelors  exposed  to  risk  of  both
marriage  and  death  at  central  age  x  +  I-

Ex+H  =  2(n;,_i  -  /x_i)  4-  Wx  -  \[ex  -\-  Wj. -\-  (bm)x  +  {bd)^]
=  S(nx_i  —  fx-i)  4-  rix  —  Ifx,
where

fz  =  Cx  -{-  Wx  -\-  {bm)x  +  ibd)x.

It  is  important  to  note  the  simplicity  of  this  method  which  re-
quires no  separation  of  both  (brnjx  and  (bd)x  into  two  parts
similar  to  Mr.  Ackland's  method  of  treating  withdrawals  and
deaths.  In  the  above  formula  it  is  evident  that  the  withdrawals,
Wx,  were  recorded  as  terminating  at  their  curtate  durations,  as
well  as  those  marrying,  dying,  and  existing.  The  accuracy  of
assuming  one  half  a  year  of  exposure  beyond  curtate  durations
for  all  terminations  should  of  course  be  tested  before  the  assump-

60  CONSTRUCTION    OF   MORTALITY   TABLES.

Mortality  tables  based  upon  the  records  of  insured  lives  which
contain  more  than  two  decremental  factors  would  rarely  be
constructed.  The  principles  involved  are  the  same  as  those
covered  by  the  preceding  discussion  of  double  decrement  tables.
The  computation  of  central  rates  will  usually  be  found  of  ad-
vantage, because  the  exposed  to  risk  has  the  same  value  for
all  central  rates  at  the  same  age,  and  from  the  central  rates  the
table  may  be  constructed  and  also  the  annual  rates  (per  year  of
exposure)  may  be  calculated.  These  latter  rates  will  often  be
wanted  in  order  that  they  may  be  compared  with  similar  rates
from  other  experiences.  It  might  also  happen  that  some  of
the  rates  entering  into  the  multiple  decrement  table  are  to  be
obtained  from  the  investigation  while  others  are  to  be  taken
from  a  different  experience.  In  such  a  case  we  must  obtain  the
central  or  annual  rates,  because  we  cannot  compute  the  prob-
ability of  a  life  surviving  and  otherwise  remaining  in  the  re-
quired status  unless  all  the  contingencies,  as  for  example,  death,
total  and  permanent  disability  and  withdrawal,  entered  into
the  single  investigation.

Chapter  VII.    Miscellaneous.

There  will  now  be  considered  various  points  in  connection  with
the  construction  of  mortality  tables  which  have  not  been  pre-
viously discussed.

Final  Series  Method:  In  compiling  the  mortality  experience  of  a
life  insurance  company,  it  will  be  found  that  while  there  may  be
many  old  policies  which  have  run  their  course,  there  will  be  a
considerable  number  existing  at  the  close  of  the  observations.
In  the  preparation  of  the  mortality  experience  of  the  Thirty
American  Life  Offices,  which  was  based  on  calendar  years  and
is  described  in  Actuarial  Studies  No.  1,  it  was  found  that  the
average  duration  of  the  policies  was  only  4.36  years,  and  that  of
982,734  male  lives  entering  into  the  investigation  527,157  were
existing  at  the  close  of  the  observations.  The  method  of  Final
Series  was  adopted  to  carry  the  existing  forward  to  their  ultimate
destination  of  death  or  withdrawal,  and  this  was  done  on  the
assumption  that  the  future  experience  on  the  existing  would
follow  the  same  select  rates  of  mortality  and  withdrawal  that
had  actually  been  experienced  in  the  past.  Incorporating  this
hypothetical  data  obviously  has  an  effect  only  on  aggregate
tables.  The  method  employed  may  best  be  illustrated  by  an
example  taken  from  page  36  of  the  volume*  giving  the  results  of
the  investigation.  The  table  on  page  62  for  one  age  at  entry
represents  amounts  to  the  nearest  thousand  with  000  omitted.

Although  the  values  given  represent  thousands  of  dollars  of
insurance  it  will  be  easier  to  discuss  them  as  though  they  repre-
sented lives  or  policies.

The  original  data  show  that  there  are  11,763  existing  at  the
end  of  year  0  (i.e.,  the  first  calendar  year  of  insurance)  to  be
carried  forward,  and  out  of  176,799  entrants  of  year  1  there  are
27,361  discontinued  and  1743  in  claims  in  that  year.  In  the
final  series  the  entrants  on  year  1  include  the  original  176,799
entrants  and  the  11,763  existing  at  the  end  of  the  previous  year,
making  a  total  of  188,562.  It  is  assumed  that  the  discontinued
and  claims  arising  out  of  the  188,582  will  be  in  proportion  to  the
discontinued  and  claims  arising  out  of  the  176,799  entrants  in

*  System  and  Tables  of  Life  Insurance — Meech,  Vol.  I.

61

62

CONSTRUCTION    OF   MORTALITY   TABLES

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FROM   THE   RECORDS   OF   INSURED   LIVES.  63

the  observed  data.  The  discontinued  and  claims  in  the  final
series  will  therefore  be  found  by  multiplying  the  amount  of
these  items  in  the  original  data  by  188,562/176,799,  namely,  by
1.0665.  This  gives  29,181  discontinued  and  1,859  in  claims,
a  total  of  31,040;  which  subtracted  from  188,562  gives  157,522,
the  entrants  on  year  2.  It  may  be  pointed  out  that  the  number
of  entrants  on  year  0  is  the  total  of  those  passing  out  of  observa-
tion as  the  existing,  discontinued  and  claims,  namely,  102,484
+  84,429  +  9,549,  or  196,462;  and  if  the  adjusted  sum  of  dis-
continued and  claims  is  deducted  each  year  the  balance  gives
the  entrants  on  the  following  year.  In  this  way  the  total  ex-
isting according  to  the  original  data  is  automatically  taken  care
of  in  the  final  series  by  merely  calculating  the  amounts  passing
out  of  observation  by  discontinuance  and  claims.

The  exposed  to  risk  of  death  after  year  0  is  obtained  by  sub-
tracting from  the  entrants  one  half  the  discontinued.  For  the
year  0  the  exposed  is  made  up  of  half  the  entrants,  less  half  the
discontinued,  or  what  is  the  same  thing,  one-half  the  difference
between  the  entrants  and  discontinued  for  that  year.  The
amount  of  claims  divided  by  the  exposed  to  risk  of  death  gives
the  qx  values.

The  exposed  to  risk  and  deaths  for  the  first  calendar  year  are
unaffected;  for  the  second  year  they  are  increased  by  a  small
percentage,  for  the  third  year  by  a  larger  percentage,  and  so  on
till  the  twenty-eighth  year  when  the  process  terminated.  The
process  as  illustrated  in  the  foregoing  schedule  resulted  in  a
multiplier  greater  than  100  for  some  ages  at  the  longest  durations,
a  figure  which  appears  to  have  alarmed  the  computers,  as  the
multipliers  finally  adopted  were  those  brought  out  by  the
original  process  multiplied  by  (1.04)~",  where  n  represents  the
number  of  the  year  of  insurance.  Even  with  this  adjustment
the  data  of  the  later  durations  would  at  some  ages  be  multiplied
twenty  and  thirty  fold.  The  hypothetical  data  for  the  aggregate
table  thus  included  an  enormously  larger  proportion  of  experi-
ence at  the  longer  durations  than  did  the  actual  material,  re-
sulting in  a  considerable  increase  in  the  rate  of  mortality,  especi-
ally at  the  older  ages.  It  is  open  to  question  whether  the
final  series  method  may  not  disturb  the  mortality  rates  to  such
an  extent  as  to  render  the  results  untrustworthy.  So  far  as  we
know  it  has  been  applied  only  in  connection  with  the  30  American
Offices'  Experience.

64  CONSTRUCTION   OF   MORTALITY  TABLES

King's  Method:  Mr.  George  King  in  Volume  XXVII,  page  218,
of  the  Journal  of  the  Institute  of  Actuaries,  described  a  method  of
ascertaining  the  mortality  experience  of  a  life  insurance  company
between  valuation  dates.  It  is  quite  customary  for  British  life
insurance  companies  to  make  a  valuation  of  policy  liabilities  and
to  distribute  surplus  once  in  five  years.  A  yearly  valuation  is
sometimes  made  for  the  guidance  of  the  companies,  but  the
valuation  on  which  the  quinquennial  distribution  of  surplus  is
based  is  made  once  in  five  years,  and  King's  method  was  de-
signed for  the  purpose  of  readily  ascertaining  the  mortality
experience  of  a  company  between  valuation  dates.

Assuming  that  the  valuation  date  is  December  31st,  and
changing  the  notation  used  by  Mr.  King,  the  method  is  as
follows:

Let  Sx  =  survivors  at  commencement  of  observations  where  x

is  the  age  nearest  birthday  on  December  Slst,

the  day  before  such  commencement;
w,  =  entrants  during  period  of  observation,  where  x  is

the  age  nearest  birthday  on  nearest  December  Slst

to  date  of  entry;
Wx  =  withdrawals,  where  x  is  age  nearest  birthday  on

nearest  December  Slst  to  date  of  withdrawal;
6x  =  deaths,  where  x  is  age  nearest  birthday  on  December

Slst  preceding  death;
Bx  =  existing  at  December  Slst  at  close  of  observations,

X  being  age  nearest  birthday  at  that  date;  and
Ex  =  exposed  to  risk  in  year  of  age  x  to  x  +  1.

Then:
and

Ex  =  Ex-i  +  Sx  -\-  Ux  -  Wx  -  6x-l  -  €x

^'  =  K

The  Sx  and  6,  are  taken  directly  from  the  classification  registers
which  are  used  in  connection  with  the  valuation  of  the  policy
liabilities  and  it  is  necessary  to  classify  merely  the  n,,  Wx  and  6x-
Furthermore,  fractional  exposures  are  avoided.  Although  in-
tended to  apply  only  for  a  short  term  of  years,  this  method  might
be  extended  to  apply  to  the  entire  mortality  experience  of  a
companj'.

FROM  THE   RECORDS   OF   INSURED   LIVES.  65

Mr.  King  points  out  that,  as  compared  with  the  Institute
Method,  his  formula  avoids  exposures  of  fractions  of  a  year  and
is  more  accurate  where  entries  and  exits  are  not  uniformly  dis-
tributed throughout  the  year.  Moreover  the  Institute  Method
gives  no  exposure  to  lives  entering  and  discontinuing  in  the  same
calendar  year,  while  approximate  allowance  is  made  for  them
under  Mr.  King's  method.  He  calls  attention  also  to  the  rapidity
of  the  calculation  of  the  exposed  to  risk  by  summation,  as  follows:

Ex  =  S(sx  -\-  Ux  —  Wx  —  dx-i  —  ex)
and  letting

Sx  -\-  Ux  =  hx
and

Wx  +  dx-i  +  ex  =  fx)
then

Ex  =  S^i  —  2/j!
=  ^{hx-fx).

Mr.  King's  method  was  intended  to  be  applied  in  British  Life
Offices  where  the  method  of  valuation  is  different  from  that
followed  in  this  country.  The  majority  of  the  business  was
tabulated  according  to  the  oflfice  age  and  was  in  convenient  form
to  apply  the  method.  If  it  were  desired  to  make  use  of  this
plan  in  an  ofl&ce  using  the  American  form  of  valuation,  the
necessary  particulars  would  require  additional  calculation.

Mr.  Whittall  has  contended  (/.  /.  A.,  XXXI,  184)  that  Mr.
King's  method  is  one  of  mean  ages.  Looking  at  the  question
graphically,

Dec.  31  Dec.  31  Dec.  31

June  30       I       June  30       I       June  30       I

an  entrant  in  the  period  he  is  relegated  to  6  and  his  nearest
birthday  at  point  6  must  lie  between  a  and  c;  an  entrant
in  the  period  cd  is  relegated  to  d  and  his  nearest  birthday  at  the
point  d  must  lie  between  c  and  e.  This  is  equivalent  to  taking
mean  ages  with  years  running  from  June  to  June  instead  of
December  to  December.  This  is  apparent  if  it  be  assumed  that
the  office  years  run  from  June  30th  to  June  30th.  Then  an
entrant  in  the  period  cd  would  be  relegated  to  c  and  his  nearest
birthday  at  that  point  would  lie  between  6  and  d,  that  is,  within
the  calendar  year.    Similarly  an  entrant  in  the  latter  half  of

66  CONSTRUCTION    OF   MORTALITY  TABLES

the  financial  year,  de,  would  be  relegated  to  e  and  at  that  point
the  nearest  birthday  would  lie  between  d  and  /,  that  is,  within
the  calendar  year.  In  the  case  of  an  office  with  a  financial
year  ending  June  30th,  the  same  results  would  be  attained  by
adopting  a  system  of  mean  ages,  i.e.,  subtracting  year  of  birth
from  year  of  entry.  The  same  principle  applies  to  the  survivors,
withdrawals,  deaths,  and  existing,  allowing  for  the  proper  in-
cidence of  each.

Mr.  King,  in  discussing  Mr.  Whittall's  paper,  disagreed  that
it  is  a  method  of  mean  ages,  because  that  method,  as  ordinarily
applied,  permits  as  much  as  a  year's  error  in  age  in  any  particular
case,  while  his  use  of  the  nearest  birthday  on  the  nearest  De-
cember 31st,  never  permitted  an  error  on  the  assumed  date  of
entry  or  exit  of  more  than  six  months.  They  both  agreed  that
it  was  difficult  to  classify  the  method  as  one  purely  of  policy
or  calendar  years.  Mr.  King,  however,  had  previously  stated
(J.  I.  A.,  Vol.  XXIX,  178)  that  "the  averages  were  so  taken  that
they  really  got  policy  years.  The  average  was  so  arranged  as  to
make  the  policy  years  harmonize  with  the  financial  years  of  the
company."

Compilation  of  Select  Tables  where  Data  have  not  been  Traced  on
Policy  Year  Basis:  A  good  example  of  the  formation  of  select
tables  from  a  calendar  year  experience  is  the  select  tables  formed
by  Dr.  Sprague  from  the  data  of  the  Mortality  Experience  of
the  Institute  of  Actuaries  (J.  I.  A.,  XXI,  p.  229).  The  data
followed  calendar  years  and  the  lives  were  taken  at  age  next
birthday  at  entry.  It  was  assumed  that  the  lives  entered  on
the  average  in  the  middle  of  the  calendar  year  and  attained  age
next  birthday  at  the  end  of  the  year.  The  lives  were  therefore
aged  X  —  ^[at  the  date  of  entry  in  the  middle  of  the  year,  x  being
age  next  birthday  at  entry.

The  values  of  ^px,  liPx,  2iPxy  etc.,  were  first  calculated  for
each  age  (x  —  1)  at  entry,  the  original  exposed  to  risk  being
modified  to  give  an  exposure  of  3  months  to  those  withdrawing
in  the  calendar  year  of  entry.  Then,  in  order  to  obtain  sufficient
data  the^experience  was  taken  in  groups  of  five  ages.  To  avoid
giving  undue  weight  to  any  particular  age,  50,000  entrants  were
taken  for  age  {x  —  2|),  100,000  for  ages  (x  —  1^),  {x  —  ^),
(^  +  Difjand  (x  -h  1^),  and  50,000  for  age  (x  -f-  2^),  making
500,000  in  each  group.  By  taking  50,000  at  the  youngest  and
oldest  age  in'^each^group  an  even  100,000  at  each  age  was  ob-

FROM   THE   RECORDS   OF   INSURED   LIVES.  67

tained,  since  these  extreme  ages  are  each  included  in  two  groups.
The  values  of  l[x-2j4]+}^,  hx-2yi]+i}4,  hx-2H]+il4f  ^^c,  were  next
calculated,  being  based  upon  the  50,000  assumed  for  the  value
of  Z[i_2j4].  Similar  values  for  each  age  at  entry  were  calculated
and  a  combination  of  the  values  for  the  same  durations  for  ages
at  issue  {x  -  2^),  (x  —  U),  (x  -  ^),  (x  +  h)>  (^  +  1^)  and
(x  4-  2^)  were  taken  as  the  values  for  the  average  age  x  at  entry,
the  value  of  l[x]  being  500,000.  The  total  deaths  at  the  end  of
^,  If,  2f,  etc.,  years,  were  then  found  and  by  an  interpolation
formula  the  total  deaths  at  the  end  of  1,  2,  3,  etc.,  years  were
calculated.  Then  by  taking  the  first  differences  of  the  results  the
numbers  dying  in  each  year  of  insurance  could  be  obtained.
Having  obtained  in  this  way  the  necessary  data  for  quinquennial
rates  of  mortality  for  the  first  insurance  year,  (6)  the  ratio  of
the  rate  of  mortality  of  the  second  insurance  year  to  that  of  the
first,  and  (c)  the  corresponding  ratio  between  the  third  and
second  insurance  years.  The  values  of  q[x]+n  for  n  =  0,  n  =  1
and  n  =  2  being  thus  available,  and  for  n  >  4  being  taken  as
those  of  Qx+n  by  the  H^^^)  Table,  values  for  n  =  3  and  n  =  4
were  supplied  by  interpolation.  Interpolation  was  used  also
to  supply  the  values  for  other  than  quinquennial  entry  ages.
In  some  cases,  for  values  of  n  less  than  5,  H^^^)  values  of  qx+n
were  substituted  for  those  found  as  above  described  because
Qx+n  by  the  H^^^)  table  was  less.

A  corresponding  method  would  have  to  be  followed  in  order  to
obtain  select  tables  from  an  experience  based  upon  life  years.

However,  when  it  is  possible  to  obtain  suflScient  data  select
tables  should  always  be  constructed  by  a  policy  year  method  and
not  by  an  indirect  method  such  as  that  employed  by  Dr.  Sprague.

Construction  of  Mortality  Tables  from  Limited  Data:  It  not
infrequently  happens  that  the  data  upon  which  a  mortality
table  is  based  are  not  sufficient  to  give  reliable,  if  indeed  any,
results  at  the  young  and  old  ages.  Where  the  data  are  graduated
by  assuming  some  law  of  mortality,  as  described  in  Actuarial
Studies  No.  4,  the  values  for  the  young  and  old  ages  may  be
supplied  by  assuming  that  the  law  holds  throughout  the  table.

When  a  table  is  not  so  graduated  it  may  be  necessary  to  sub-
stitute for  certain  portions  of  the  table  which  is  being  prepared,
the  data  from  some  standard  table  which  are  believed  to  approxi-
mate to  the  mortality  of  the  class  of  lives  on  which  the  new  table

68  CONSTRUCTION   OF   MORTALITY  TABLES

is  based.  The  junction  of  the  data  of  the  standard  table  with
that  of  the  new  table  may  be  smoothed  off  by  graduation  or
interpolation.

Where  the  data  are  very  limited  throughout,  the  actual  deaths
may  be  compared  with  the  expected  deaths  according  to  some
standard  table,  grouping  a  sufficient  number  of  ages  to  secure
enough  data  to  give  a  general  idea  of  the  ratio  of  the  actual
mortality  to  that  shown  by  the  standard  table  at  various  ages.
The  ratios  for  intermediate  ages  may  then  be  found  by  interpola-
tion and  the  new  table  formed  by  applying  the  adjusted  ratios
to  the  data  of  the  standard  table.

Homogeneity  of  Data:  In  constructing  a  mortality  table  our
purpose  is  usually  not  only  to  ifind  out  what  rates  of  mortality
have  been  experienced  among  the  lives  investigated  but  also
to  have  a  table  which  can  be  applied  to  estimate  future  events.
The  latter  purpose  is  the  chief  one  in  practically  all  cases.  If
we  are  to  be  able  to  apply  a  table  with  any  confidence  we  must
know  the  character  of  the  lives  that  contributed  the  experience
and  any  special  conditions  to  which  their  mortality  was  subject.
It  therefore  becomes  obvious  that  our  material  must  be  suffi-
ciently homogeneous  so  that  the  results  can  be  said  to  apply  to  a
reasonably  stable  group  of  lives.

A  little  consideration,  however,  will  show  that  when  we  speak
of  "homogeneous  data,"  we  are  using  a  relative  term.  It  is
impossible  to  imagine  a  large  group  of  lives  which  is  not  made  up
of  many  subgroups  each  of  which  is  subject  to  slightly  different
mortality  rates.  For  example  in  insurance  experience,  those
lives  which  are  insured  under  plans  calling  for  the  lowest  premium
rates  will,  generally  speaking,  show  the  highest  rates  of  mortality.
Where  those  insuring  have  a  free  choice  as  to  whether  their
insurance  shall  be  upon  the  participating  or  non-participating
plan,  it  will  generally  be  found  that  those  selecting  non-participat-
ing policies  will  show  the  higher  mortality,  though  there  is  not
the  same  reason  for  supposing  that  a  company  issuing  only  non-
participating  policies  will  experience  a  higher  mortality  than
another  company  granting  insurance  only  upon  the  participating
plan.  The  relative  mortality  of  male  and  female  lives  differs
at  various  ages.  The  residents  of  different  sections  of  the
United  States  are  subject  to  different  rates  of  mortality.  Even
if  we  disregard  outside  influences  we  shall  find  degrees  of  difference
among  lives  insured  in  one  company  at  standard  rates  due  to

FROM   THE   RECORDS   OF    INSURED   LIVES.  69

physical  or  occupational  causes.  Then  again  mortality  rates
will  be  influenced  by  such  external  causes  as  agency  methods,
science  and  sanitation  and  in  the  science  of  selection.

Mr.  G.  F.  Hardy*  has  given  the  following  statement  of  the
problem  and  its  proper  solution.

"The  Actuary  constructs  tables  not  merely  to  show  what  has  happened  in
the  past,  but  to  enable  him  to  forecast  the  future,  and  as  he  requires  these
tables  as  a  basis  for  financial  operations,  considerations  are  introduced  which
do  not  arise  in  the  treatment  of  purely  statistical  tables.  Whatever  class
of  events  the  Actuary  may  have  to  deal  with,  will  be  subject  to  change  with
the  lapse  of  time.  That  portion  of  the  class  he  has  been  able  to  observe  hes
necessarily  in  the  past;  the  conclusions  he  has  derived  from  their  study  he
proposes  to  extend  to  the  future.  He  must  therefore  consider  how  far  the
observed  characters  of  the  class  are  changing  or  permanent,  and  must  en-
deavour to  distinguish  between  changes  representing  permanent  tendencies
and  those  due  merely  to  temporary  fluctuations.  In  the  selection  of  data
suitable  for  his  purpose  the  Actuary  will  aim  on  the  one  hand  at  a  sufficiently
broad  basis  both  in  space  and  time  to  eliminate  the  effects  of  local  and  tem-
porary fluctuations,  and  on  the  other  hand  he  will  aim  at  obtaining  as  far  as
possible  a  homogeneous  group  of  data.  These  two  aims  are  more  or  less  in
conflict,  and  he  will  lean  to  the  one  side  or  the  other,  according  to  the  object
he  has  in  view.  Where,  for  example,  that  object  is  to  produce  a  table  that  may
be  adopted  as  a  general  standard  by  various  institutions,  often  differing  con-
siderably as  to  their  individual  experience,  he  must  aim  at  a  correspondingly
broad  foimdation.  In  these  circumstances  it  will  not  generally  be  possible
to  obtain  a  really  homogeneous  experience.  If  it  is  a  question  of  the  mortaUty
of  assured  lives,  for  instance,  this  will  be  found  to  be  affected  by  endless  in-
dividual variations,  age,  sex,  duration  of  assurance,  occupation,  civil  condi-
tion, class  of  assurance,  character  of  the  insuring  office,  etc.,  etc.,  and  from
such  material  approximately  homogeneous  data  could  only  be  obtained  by
cutting  up  the  experience  into  comparatively  small  groups  and  thus  sacrificing
all  generahty.  This  can  be  avoided  in  practice  by  first  excluding  all  extreme
variations.  The  sexes  will  be  separately  treated,  lives  so  impaired  as  to
prospects  of  longevity  by  personal  health,  family  history,  occupation,  or
residence  in  unhealthy  districts  as  to  be  "rated  up"  will  be  excluded,  as  also
classes  of  assurance  that  may  be  supposed  subject  to  rates  of  mortality  dif-
fering from  the  average.  When  the  data  has  thus  been  trimmed  of  the  ex-
treme variations,  a  body  of  experience  will  generally  remain  not  greatly
shrunken  from  its  original  dimensions  and  in  which  the  discontinuous  varia-
tions are  sufficiently  numerous  and  individually  unimportant  to  render  the
data  for  practical  purposes  homogeneous.  The  rates  of  mortality,  or  with-
drawal, can  then  be  treated  as  functions  of  the  two  remaining  variables  of
importance,  the  age  and  the  time  elapsed  from  date  of  entry;  or  as  functions
of  the  age  only  from  the  point  at  which  the  factor  of  duration  may  be  found
to  be  unimportant."

*  "The  Theory  of  the  Construction  of  Tables  of  MortaUty,  etc.,"  p.  16.

70  CONSTRUCTION    OF   MORTALITY   TABLES

Spurious  Selection:  After  a  reasonably  homogeneous  set  of
lives  have  been  taken  to  form  select  and  ultimate  tables  there  is
still  some  question  whether  the  effect  of  the  selection  of  risks
can  be  measured  precisely  by  the  difference  between  the  ultimate
rates  and  the  select  rates  for  the  same  attained  age  but  for  dura-
tions within  the  select  period ;  nor  can  we  be  altogether  sure  that
the  effect  of  selection  is  felt  for  the  exact  period  shown  by  the
tables.  In  the  first  place  it  will  be  noted  that  the  practical
period  of  selection  at  all  ages  at  entry  contrary  to  the  evidence
of  the  unadjusted  figures,  and  may  also  result  in  an  appreciable
change  in  the  select  rates  for  the  later  part  of  the  period  at  some
ages  in  order  to  form  a  smooth  junction  with  the  ultimate  rates.

Mr.  W.  P.  Elderton  introduced  the  term  "Spurious  Selection"
(J.  /.  A.,  Vol.  XL,  221)  to  describe  "the  selection  indicated  by  a
difference  in  q  which  has  arisen  entirely  from  statistical  proc-
esses." He  examined  the  case  where  a  table  is  formed  by
amalgamating  two  classes  of  data  to  which  he  applied  the  symbols
Eix]+t  and  E'ix]+t,  the  prime  indicating  the  data  subject  to  the
heavier  mortality  rate.  Using  q^+t  and  q'x+t  to  indicate  the
respective  ultimate  rates  of  mortality  to  which  the  classes  are
subject,  the  rate  of  mortality  that  would  be  found  for  the
{t  +  l)th  year  from  the  amalgamated  data  would  be

Eix]+tqx+t  +  E  ix]+tq  x+t
E[x]+t  +  E'ix]+t

Similarly  the  rate  for  the  same  attained  age  arising  from  the
next  year  of  duration  is

Eix-i]+t+iqx+t  -\-  E  ix-i]+t+iq  x+t
Eix-i]+t+i  +  E  ix-\]+t+i

The  rates  qx+t  and  q'x+t  are  used  in  both  cases  since  each  class  of
data  is  assumed  to  have  passed  beyond  its  period  of  selection.
It  is  seen,  however,  that  if  q  and  q'  are  unequal  the  resulting
mortality  rates  for  the  amalgamated  table  will  be  unequal  though
they  apply  to  the  same  attained  age.  If  the  latter  expression
is  the  greater  then  selection  will  appear  to  be  still  effective
whereas  we  know  it  has  really  ceased,  but  if  the  former  is  the
greater  we  shall  get  the  opposite  effect  to  that  of  selection,  or
what  may  be  called  a  tendency  to  conceal  the  effects  of  selection.

FROM  THE   RECORDS   OF  INSURED    LIVES.  71

The  analysis  may  be  carried  further,  for

Eix]+tq  +  E\x]+tq'  >  E[x-i]+i+iq  +  E'[x-iu-t+iq'
Eix]+t  +  E'ix]+t    "^    E[x-i]+t+i  +  E\x.i]+t+i
as
^lx-i]-i-t+iE\x]+tq'  4-  E[x]+tE\x-i]i-t+iq

=  E\x..i]+t+iE[x]+tq'  -\-  Eix-i]+t+iE\x]+iq
or  as

Eix-i]+i+iE'ix]+t{q'  —  q)  ^  E[x]+tE'ix-i]+t+i(q'  —  q)
or  as

E[x]  +  t  "^    ^[x-l]  +  (+l  ■

In  other  words  we  shall  get  the  appearance  of  selection  that
does  not  exist  if  the  right-hand  member  of  the  last  equation  is  the
greater,  i.e.,  if  the  proportion  of  data  subject  to  the  heavier
mortality  is  increasing  with  duration  for  the  same  attained  age.
We  may  come  to  this  conclusion  also  from  general  reasoning,
for  in  such  a  case  the  ultimate  data  will  contain  an  undue  amount
of  the  heavier  mortality  data  while  the  very  early  years  of  dura-
tion will  be  largely  made  up  of  the  lower  mortality  data,  so  that
in  order  to  have  the  mortality  rates  pass  from  the  low  initial
select  rates  to  the  heavy  ultimate  rates  will  require  us  to  pass
through  an  increased  number  of  years  of  duration.  Thus  the
apparent  effect  of  selection  is  increased  both  by  increasing  the
differences  between  the  select  and  ultimate  rates  and  also  by
increasing  the  period  of  selection.

If,  on  the  other  hand,  the  left-hand  member  of  this  equation
is  the  greater,  we  obtain  the  contrary  effect,  our  initial  select
rates  being  increased  by  the  heavier  mortality  and  the  ultimate
rates  being  lowered  by  the  larger  proportion  of  lower  mortality,
thus  tending  to  offset  the  true  effects  of  selection.

The  former  condition  would  seem  to  be  the  more  likely  one  in  a
life  insurance  experience  because  of  the  fact  that,  generally,
there  has  been  evidence  that  mortality  has  decreased  with  ad-
vancing calendar  year  of  entry.  The  longer  the  duration  the
larger  would  be  the  proportion  of  data  arising  from  early  issue,
if  the  investigation  is  terminated  for  all  policies  on  a  certain
date  or  in  a  certain  year.

Mr.  P.  C.  H.  Papps  investigated  another  special  case  {T.  A.  S.
A.,  Vol.  XIII,  211)  in  which  spurious  selection  may  be  due  to  a
progressive  improvement  in  mortality  with  advancing  calendar

72  CONSTRUCTION    OF  MORTALITY  TABLES.

years  of  exposure  affecting  all  the  data  irrespective  of  year  of
issue.  Such  an  effect  might  be  due  to  improvement  in  the  general
health  of  the  population.  He  assumed  further  that  the  data
included  only  policies  issued  during  the  period  of  investigation.
He  has  shown  that,  for  an  equal  volume  of  issues  each  year,  the
result  would  be  to  produce  too  low  an  effect  of  selection.

This  result  could  also  be  reached  by  general  reasoning  for  the
mortality  rate  for  the  longest  year  of  duration  would  be  deter-
mined from  exposures  in  the  last  year  of  the  investigation,  while
the  rate  for  the  first  policy  year  would  come  from  exposures
over  the  entire  period,  with  intermediate  durations  passing  from
one  extreme  to  the  other.  Consequently  the  mortality  rates
for  the  early  years  of  duration  would  be  relatively  high  com-
pared with  those  of  the  later,  or  ultimate  years.

As  pointed  out  by  Mr.  F.  H.  Johnston,  in  discussing  Mr.
Papps's  paper,  this  effect  vanishes  when  a  table  is  constructed
from  data  that  includes  the  experience  of  issues  prior  to  the
period  of  investigation,  assuming  that  the  distribution  of  the
exposed  to  risk  for  any  one  policy  year  was  chronologically  the
same  as  for  any  other  policy  year — which  is  fairly  close  to  the
fact.  In  such  a  case  the  proportions  of  the  exposed  to  risk  at
duration  t  arising  in  the  first,  second,  third,  etc.,  years  of  the
investigation  is  the  same  as  the  proportions  for  other  durations.*
Then  the  mortality  rate  for  each  year  of  duration  would  be
equal  to  a  constant  multiplied  by  the  rate  for  that  duration  for
any  specified  year  of  the  investigation.  This  would  also  be
true  if  the  mortality  rates  changed  in  any  manner  from  year  to
year  of  the  investigation.

The  question  whether  spurious  selection  exists  is  a  complex  one
in  any  case,  and  its  importance  in  any  table  constructed  from
carefully  selected  data  in  the  usual  way  is  probably  small.  It
does  emphasize,  however,  the  need  for  obtaining  homogeneous
data  if  we  are  to  place  much  reliance  on  the  effect  of  selection
displayed  by  the  resulting  tables.

*  This  woxild  not  be  true  in  general  for  the  longest  durations,  but  this  would
have  little  e£fect  on  the  ultimate  rates  in  most  cases.

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